The theory of dynamical systems plays an important role in the mathematical description of time-dependent processes in various fields, such as physics and technology, biology or economics. It includes the study of systems of ordinary differential equations, partial differential equations, delay-differential equations and iterated mappings.

The research in this field at WIAS is mainly focussed to develop the mathematical background for problems from the dynamics of semiconductor lasers and nonlinear optics. This leads to the following topics:

  • theory of singularly perturbed systems and asymptotic methods
  • bifurcation theory and numerical bifurcation analysis by path-following methods
  • dynamics of delay-differential equations
  • collective dynamics in large coupled systems
  • self-organization and control of spatio-temporal patterns

An important topic in the field of delay-differential equations related to models of lasers with optical feedback is the limit of large delay time. Significant mathematical problems are asymptotic descriptions of the spectra for equilibria and periodic solutions.

A further major focus are high-dimensional effects and complex dynamics in large coupled systems. Here in particular collective dynamics and pattern formation in systems of coupled oscillators are of interest.

Coherence-incoherence pattern in a two-dimensional array of coupled phase oscillators
Fig1. -Coherence-incoherence pattern in a two-dimensional array of coupled phase oscillators

Publications

  Monographs

  • M. Tlidi, R. Lefever, A.G. Vladimirov, On Vegetation Clustering, Localized Bare Soil Spots and Fairy Circles, in: Dissipative Solitons: From Optics to Biology and Medicine, N. Akhmediev, A. Ankiewicz, eds., 751 of Lecture Notes in Physics, Springer, Berlin, Heidelberg, 2008, pp. 381-402, (Chapter Published).

  Articles in Refereed Journals

  • A. Pimenov, S. Slepneva, G. Huyet, A.G. Vladimirov, Dispersive time-delay dynamical systems, Physical Review Letters, 118 (2017) pp. 193901, DOI 10.20347/WIAS.PREPRINT.2324 .
    Abstract
    We present a theoretical approach to model the dynamics of a dispersive nonlinear system using a set of delay differential equations with distributed delay term. We illustrate the use of this approach by considering a frequency swept laser comprimising a semiconductor optical amplifier (SOA), a tunable bandpass filter and a long dispersive fiber delay line. We demonstrate that this system exhibits a rich spectrum of dynamical behaviors which are in agreement with the experimental observations. In particular, the multimode modulational instability observed experimentally in the laser in the anomalous dispersion regime and leading to a turbulent laser output was found analytically in the limit of large delay time.

  • M. Hintermüller, C.N. Rautenberg, On the uniqueness and numerical approximation of solutions to certain parabolic quasi-variational inequalities, Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society, 74 (2017) pp. 1--35.
    Abstract
    A class of abstract nonlinear evolution quasi-variational inequality (QVI) problems in function space is considered. The abstract framework developed in this paper includes constraint sets of obstacle and gradient type. The paper address the existence, uniqueness and approximation of solutions when the constraint set mapping is of a special form. Uniqueness is addressed through contractive behavior of a nonlinear mapping whose fixed points are solutions to the QVI. An axiomatic semi-discrete approximation scheme is developed, which is proven to be convergent and which is numerically implemented. The paper ends by a report on numerical tests for several nonlinear constraints of gradient-type.

  • O. Omel'chenko, L. Recke, V. Butuzov, N. Nefedov, Time-periodic boundary layer solutions to singularly perturbed parabolic problems, Journal of Differential Equations, (2017) pp. 2--40, DOI 10.20347/WIAS.PREPRINT.2300 .
    Abstract
    In this paper, we present a result of implicit function theorem type, which was designed for application to singularly perturbed problems. This result is based on fixed point iterations for contractive mappings, in particular, no monotonicity or sign preservation properties are needed. Then we apply our abstract result to time-periodic boundary layer solutions (which are allowed to be non-monotone with respect to the space variable) in semilinear parabolic problems with two independent singular perturbation parameters. We prove existence and local uniqueness of those solutions, and estimate their distance to certain approximate solutions.

  • A. Pimenov, D. Rachinskii, Robust homoclinic orbits in planar systems with Preisach hysteresis operator, Journal of Physics: Conference Series, 727 (2016) pp. 012012/1--012012/15.
    Abstract
    We construct examples of robust homoclinic orbits for systems of ordinary differential equations coupled with the Preisach hysteresis operator. Existence of such orbits is demonstrated for the first time. We discuss a generic mechanism that creates robust homoclinic orbits and a method for finding them. An example of a homoclinic orbit in a population dynamics model with hysteretic response of the prey to variations of the predator is studied numerically

  • I. Omelchenko, O. Omel'chenko, A. Zakharova, M. Wolfrum, E. Schöll, Tweezers for chimeras in small networks, Physical Review Letters, 116 (2016) pp. 114101/1--114101/5.
    Abstract
    We propose a control scheme which can stabilize and fix the position of chimera states in small networks. Chimeras consist of coexisting domains of spatially coherent and incoherent dynamics in systems of nonlocally coupled identical oscillators. Chimera states are generically difficult to observe in small networks due to their short lifetime and erratic drifting of the spatial position of the incoherent domain. The control scheme, like a tweezer, might be useful in experiments, where usually only small networks can be realized.

  • D. Davino, P. Krejčí, A. Pimenov, D. Rachinskii, C. Visone, Analysis of an operator-differential model for magnetostrictive energy harvesting, Communications in Nonlinear Science and Numerical Simulation, 39 (2016) pp. 504--519.
    Abstract
    We present a model of, and analysis of an optimization problem for, a magnetostrictive harvesting device which converts mechanical energy of the repetitive process such as vibrations of the smart material to electrical energy that is then supplied to an electric load. The model combines a lumped differential equation for a simple electronic circuit with an operator model for the complex constitutive law of the magnetostrictive material. The operator based on the formalism of the phenomenological Preisach model describes nonlinear saturation effects and hysteresis losses typical of magnetostrictive materials in a thermodynamically consistent fashion. We prove well-posedness of the full operator-differential system and establish global asymptotic stability of the periodic regime under periodic mechanical forcing that represents mechanical vibrations due to varying environmental conditions. Then we show the existence of an optimal solution for the problem of maximization of the output power with respect to a set of controllable parameters (for the periodically forced system). Analytical results are illustrated with numerical examples of an optimal solution.

  • K. Panajotov, D. Puzyrev, A.G. Vladimirov, S.V. Gurevich, M. Tlidi, Impact of time-delayed feedback on spatiotemporal dynamics in the Lugiato--Lefever model, Physical Review A, 93 (2016) pp. 043835/1--043835/7.
    Abstract
    We analyze the impact of delayed optical feedback (OF) on the spatiotemporal dynamics of the Lugiato-Lefever model. First, we carry out linear stability analysis and reveal the role of the OF strength and phase on the shape of the bistable curve as well as on Turing, Andronov-Hopf, and traveling-wave instability regions. Further, we demonstrate how the OF impacts the spatial dynamics by shifting the regions with different spatial eigenvalue spectra. In addition, we reveal a clustering behavior of cavity solitons as a function of the OF strength at fixed OF phase. Depending on the feedback parameters, OF can also induce a drift bifurcation of a stationary cavity soliton, as well as an Andronov-Hopf bifurcation of a drifting soliton. We present an analytical expression for the threshold of the drift bifurcation and show that above a certain value of the OF strength the system enters a region of spatiotemporal chaos.

  • D. Puzyrev, A.G. Vladimirov, S.V. Gurevich, S. Yanchuk, Modulational instability and zigzagging of dissipative solitons induced by delayed feedback, Physical Review A, 93 (2016) pp. 041801/1--041801/5.
    Abstract
    We report a destabilization mechanism of localized solutions in spatially extended systems which is induced by delayed feedback. Considering a model of a wide-aperture laser with a saturable absorber and delayed optical feedback, we demonstrate the appearance of multiple coexistent laser cavity solitons. We show that at large delays apart from the drift and phase instabilities the soliton can exhibit a delay-induced modulational instability associated with the translational neutral mode. The combination of drift and modulational instabilities produces a zigzagging motion of the solitons, which are either periodic, with the period close to the delay time, or chaotic, with low-frequency fluctuations in the direction of the soliton motion. The same type of modulational instability is demonstrated for localized solutions of the cubic-quintic complex Ginzburg-Landau equation.

  • O. Omel'chenko, M. Wolfrum, Is there an impact of small phase lags in the Kuramoto model?, Chaos. An Interdisciplinary Journal of Nonlinear Science, 26 (2016) pp. 094806/1--094806/6.
    Abstract
    We discuss the influence of small phase lags on the synchronization transitions in the Kuramoto model for a large inhomogeneous population of globally coupled phase oscillators. Without a phase lag, all unimodal distributions of the natural frequencies give rise to a classical synchronization scenario, where above the onset of synchrony at the Kuramoto threshold there is an increasing synchrony for increasing coupling strength. We show that already for arbitrarily small phase lags there are certain unimodal distributions of natural frequencies such that for increasing coupling strength synchrony may decrease and even complete incoherence may regain stability. Moreover, our example allows a qualitative understanding of the mechanism for such non-universal synchronization transitions.

  • M. Wolfrum, S. Gurevich, O. Omel'chenko, Turbulence in the Ott--Antonsen equation for arrays of coupled phase oscillators, Nonlinearity, 29 (2016) pp. 257--270.
    Abstract
    In this paper we study the transition to synchrony in an one-dimensional array of oscillators with non-local coupling. For its description in the continuum limit of a large number of phase oscillators, we use a corresponding Ott-Antonsen equation, which is an integro-differential equation for the evolution of the macroscopic profiles of the local mean field. Recently, it has been reported that in the spatially extended case at the synchronization threshold there appear partially coherent plane waves with different wave numbers, which are organized in the well-known Eckhaus scenario. In this paper, we show that for Kuramoto-Sakaguchi phase oscillators the phase lag parameter in the interaction function can induce a Benjamin-Feir type instability of the partially coherent plane waves. The emerging collective macroscopic chaos appears as an intermediate stage between complete incoherence and stable partially coherent plane waves. We give an analytic treatment of the Benjamin-Feir instability and its onset in a codimension-two bifurcation in the Ott-Antonsen equation as well as a numerical study of the transition from phase turbulence to amplitude turbulence inside the Benjamin-Feir unstable region.

  • M. Kantner, E. Schöll, S. Yanchuk, Delay-induced patterns in a two-dimensional lattice of coupled oscillators, Scientific Reports, 5 (2015) pp. 8522/1--8522/9.
    Abstract
    We show how a variety of stable spatio-temporal periodic patterns can be created in 2D-lattices of coupled oscillators with non-homogeneous coupling delays. The results are illustrated using the FitzHugh-Nagumo coupled neurons as well as coupled limit cycle (Stuart-Landau) oscillators. A "hybrid dispersion relation" is introduced, which describes the stability of the patterns in spatially extended systems with large time-delay.

  • L. Lücken, J.P. Pade, K. Knauer, Classification of coupled dynamical systems with multiple delays: Finding the minimal number of delays, SIAM Journal on Applied Dynamical Systems, 14 (2015) pp. 286--304.
    Abstract
    In this article we study networks of coupled dynamical systems with time-delayed connections. If two such networks hold different delays on the connections it is in general possible that they exhibit different dynamical behavior as well. We prove that for particular sets of delays this is not the case. To this aim we introduce a componentwise timeshift transformation (CTT) which allows to classify systems which possess equivalent dynamics, though possibly different sets of connection delays. In particular, we show for a large class of semiflows (including the case of delay differential equations) that the stability of attractors is invariant under this transformation. Moreover we show that each equivalence class which is mediated by the CTT possesses a representative system in which the number of different delays is not larger than the cycle space dimension of the underlying graph. We conclude that the 'true' dimension of the corresponding parameter space of delays is in general smaller than it appears at first glance.

  • E. Meca Álvarez, I. Mercader, L. Ramirez-Piscina, Transitions between symmetric and nonsymmetric regimes in binary-mixture convection, Physica D. Nonlinear Phenomena, 303 (2015) pp. 39--49.

  • A. Pimenov, T.C. Kelly, A. Korobeinikov, J.A. O'Callaghan, D. Rachinskii, Adaptive behaviour and multiple equilibrium states in a predator-prey model, Theoretical Population Biology, 101 (2015) pp. 24--30.
    Abstract
    There is evidence that multiple stable equilibrium states are possible in real-life ecological systems. In order to verify a hypothesis that such a multitude of equilibrium states can be caused by adapting of animal behaviour to changes of environmental conditions, we consider a simple predator-prey model where prey changes a mode of behaviour in response to the pressure of predation. This model exhibits two stable coexisting equilibrium states with basins of attraction separated by a separatrix of a saddle point.

  • S. Yanchuk, L. Lücken, M. Wolfrum, A. Mielke, Spectrum and amplitude equations for scalar delay-differential equations with large delay, Discrete and Continuous Dynamical Systems, 35 (2015) pp. 537--553.
    Abstract
    The subject of the paper are scalar delay-differential equations with large delay. Firstly, we describe the asymptotic properties of the spectrum of linear equations. Using these properties, we classify possible types of destabilization of steady states. In the limit of large delay, this classification is similar to the one for parabolic partial differential equations. We present a derivation and error estimates for amplitude equations, which describe universally the local behavior of scalar delay-differential equations close to the destabilization threshold.

  • S. Yanchuk, P. Perlikowski, M. Wolfrum, A. Stefański, T. Kapitaniak, Amplitude equations for collective spatio-temporal dynamics in arrays of coupled systems, Chaos. An Interdisciplinary Journal of Nonlinear Science, 25 (2015) pp. 033113/1--033113/8.
    Abstract
    We study the coupling induced destabilization in an array of identical oscillators coupled in a ring structure where the number of oscillators in the ring is large. The coupling structure includes different types of interactions with several next neighbors. We derive an amplitude equation of Ginzburg-Landau type, which describes the destabilization of a uniform stationary state and close-by solutions in the limit of a large number of nodes. Studying numerically an example of unidirectionally coupled Duffing oscillators, we observe a coupling induced transition to collective spatio-temporal chaos, which can be understood using the derived amplitude equations.

  • T. Jonsson, S. Berg, M. Emmerson, A. Pimenov, The context dependency of species keystone status during food web disassembly, Food Webs, 5 (2015) pp. 1--10.

  • V. Klinshov, L. Lücken, D. Shchapin, V. Nekorkin, S. Yanchuk, Multistable jittering in oscillators with pulsatile delayed feedback, Physical Review Letters, 114 (2015) pp. 178103/1--178103/5.
    Abstract
    Oscillatory systems with time-delayed pulsatile feedback appear in various applied and theoretical research areas, and received a growing interest in recent years. For such systems, we report a remarkable scenario of destabilization of a periodic regular spiking regime. At the bifurcation point numerous regimes with non-equal interspike intervals emerge. We show that the number of the emerging, so-called ``jittering'' regimes grows emphexponentially with the delay value. Although this appears as highly degenerate from a dynamical systems viewpoint, the ``multi-jitter'' bifurcation occurs robustly in a large class of systems. We observe it not only in a paradigmatic phase-reduced model, but also in a simulated Hodgkin-Huxley neuron model and in an experiment with an electronic circuit.

  • V. Klinshov, L. Lücken, D. Shchapin, V. Nekorkin, S. Yanchuk, Emergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 92 (2015) pp. 042914/1--042914/15.
    Abstract
    Interaction via pulses is common in many natural systems, especially neuronal. In this article we study one of the simplest possible systems with pulse interaction: a phase oscillator with delayed pulsatile feedback. When the oscillator reaches a specific state, it emits a pulse, which returns after propagating through a delay line. The impact of an incoming pulse is described by the oscillator's phase reset curve (PRC). In such a system we discover an unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic regular spiking solution bifurcates with several multipliers crossing the unit circle at the same parameter value. The number of such critical multipliers increases linearly with the delay and thus may be arbitrary large. This bifurcation is accompanied by the emergence of numerous ``jittering'' regimes with non-equal interspike intervals (ISIs). The number of the emergent solutions increases exponentially with the delay. We describe the combinatorial mechanism that underlies the emergence of such a variety of solutions. In particular, we show how each periodic solution consisting of different ISIs implies the appearance of multiple other solutions obtained by rearranging of these ISIs. We show that the theoretical results for phase oscillators accurately predict the behavior of an experimentally implemented electronic oscillator with pulsatile feedback.

  • V.Z. Tronciu, M. Radziunas, Ch. Kürbis, H. Wenzel, A. Wicht, Numerical and experimental investigations of micro-integrated external cavity diode lasers, Optical and Quantum Electronics, 47 (2015) pp. 1459--1464.

  • O. Omel'chenko, L. Recke, Existence, local uniqueness and asymptotic approximation of spike solutions to singularly perturbed elliptic problems, Hiroshima Mathematical Journal, 45 (2015) pp. 35--89.
    Abstract
    This paper concerns general singularly perturbed second order semilinear elliptic equations on bounded domains $Omega subset R^n$ with nonlinear natural boundary conditions. The equations are not necessarily of variational type. We describe an algorithm to construct sequences of approximate spike solutions, we prove existence and local uniqueness of exact spike solutions close to the approximate ones (using an Implicit Function Theorem type result), and we estimate the distance between the approximate and the exact solutions. Here ''spike solution'' means that there exists a point in $Omega$ such that the solution has a spike-like shape in a vicinity of such point and that the solution is approximately zero away from this point. The spike shape is not radially symmetric in general and may change sign.

  • M. Radziunas, V.Z. Tronciu, E. Luvsandamdin, Ch. Kürbis, A. Wicht, H. Wenzel, Study of micro-integrated external-cavity diode lasers: Simulations, analysis and experiments, IEEE J. Quantum Electron., 51 (2015) pp. 2000408/1--2000408/8.
    Abstract
    This paper reports the results of numerical and experimental investigations of the dynamics of an external cavity diode laser device composed of a semiconductor laser and a distant Bragg grating, which provides an optical feedback. Due to the influence of the feedback, this system can operate at different dynamic regimes. The traveling wave model is used for simulations and analysis of the nonlinear dynamics in the considered laser device. Based on this model, a detailed analysis of the optical modes is performed, and the stability of the stationary states is discussed. It is shown, that the results obtained from the simulation and analysis of the device are in good agreement with experimental findings.

  • M. Wolfrum, O. Omel'chenko, J. Sieber, Regular and irregular patterns of self-localized excitation in arrays of coupled phase oscillators, Chaos. An Interdisciplinary Journal of Nonlinear Science, 25 (2015) pp. 053113/1--053113/7.
    Abstract
    We study a system of phase oscillators with non-local coupling in a ring that supports self-organized patterns of coherence and incoherence, called chimera states. Introducing a global feedback loop, connecting the phase lag to the order parameter, we can observe chimera states also for systems with a small number of oscillators. Numerical simulations show a huge variety of regular and irregular patterns composed of localized phase slipping events of single oscillators. Using methods of classical finite dimensional chaos and bifurcation theory, we can identify the emergence of chaotic chimera states as a result of transitions to chaos via period doubling cascades, torus breakup, and intermittency. We can explain the observed phenomena by a mechanism of self-modulated excitability in a discrete excitable medium.

  • R.M. Arkhipov, I. Babushkin, M.K. Lebedev, Y.A. Tolmachev, M.V. Arkhipov, Transient Cherenkov radiation from an inhomogeneous string excited by an ultrashort laser pulse at superluminal velocity, Physical Review A, 89 (2014) pp. 043811/1--043811/10.
    Abstract
    An optical response of one-dimensional string made of dipoles with a periodically varying density excited by a spot of light moving along the string at the superluminal (subluminal) velocity is studied. We consider in details the spectral and temporal dynamics of the Cherenkov radiation, which occurs in such system in the transient regime. We point out the resonance character of radiation and the appearance of a new Doppler-like frequency in the spectrum of the transient Cherenkov radiation. Possible applications of the effect as well as different string topologies are discussed

  • A. Pimenov, T. Habruseva, D. Rachinskii, S.P. Hegarty, H. Guillaume, A.G. Vladimirov, Effect of dynamical instability on timing jitter in passively mode-locked quantum-dot lasers, Optics Letters, 39 (2014) pp. 6815--6818.
    Abstract
    We study the effect of noise on the dynamics of passively mode-locked semiconductor lasers both experimentally and theoretically. A method combining analytical and numeri- cal approaches for estimation of pulse timing jitter is proposed. We investigate how the presence of dynamical features such as wavelength bistability affects timing jitter.

  • A. Pimenov, D. Rachinskii, Homoclinic orbits in a two-patch predator-prey model with Preisach hysteresis operator, Mathematica Bohemica, 139 (2014) pp. 285--298.
    Abstract
    Systems of operator-differential equations which hysteresis operators can have unstable equilibrium points with an open basin of attraction. In this paper, a numerical example of a robust homoclinic loop is presented for the first time in a population dynamics model with hysteretic response of prey to variations of predator. A mechanism creating this homoclinic trajectory is discussed.

  • A. Pimenov, D. Rachinskii, Robust homoclinic orbits in planar systems with Preisach hysteresis operator, Journal of Physics: Conference Series, 727 (2016) pp. 012012/1--012012/15, DOI 10.1088/1742-6596/727/1/012012 .
    Abstract
    We construct examples of robust homoclinic orbits for systems of ordinary differential equations coupled with the Preisach hysteresis operator. Existence of such orbits is demonstrated for the first time. We discuss a generic mechanism that creates robust homoclinic orbits and a method for finding them. An example of a homoclinic orbit in a population dynamics model with hysteretic response of the prey to variations of the predator is studied numerically

  • M. Radszuweit, H. Engel, M. Bär, An active poroelastic model for mechanochemical patterns in protoplasmic droplets of Physarum polycephalum, PLOS ONE, 9 (2014) pp. e99220/1--e99220/15.
    Abstract
    Motivated by recent experimental studies, we derive and analyze a two-dimensional model for the contraction patterns observed in protoplasmic droplets of Physarum polycephalum. The model couples a description of an active poroelastic two-phase medium with equations describing the spatiotemporal dynamics of the intracellular free calcium concentration. The poroelastic medium is assumed to consist of an active viscoelastic solid representing the cytoskeleton and a viscous fluid describing the cytosol. The equations for the poroelastic medium are obtained from continuum force balance and include the relevant mechanical fields and an incompressibility condition for the two-phase medium. The reaction-diffusion equations for the calcium dynamics in the protoplasm of Physarum are extended by advective transport due to the flow of the cytosol generated by mechanical stress. Moreover, we assume that the active tension in the solid cytoskeleton is regulated by the calcium concentration in the fluid phase at the same location, which introduces a mechanochemical coupling.
    A linear stability analysis of the homogeneous state without deformation and cytosolic flows exhibits an oscillatory Turing instability for a large enough mechanochemical coupling strength. Numerical simulations of the model equations reproduce a large variety of wave patterns, including traveling and standing waves, turbulent patterns, rotating spirals and antiphase oscillations in line with experimental observations of contraction patterns in the protoplasmic droplets.

  • S. Slepneva, B. O'Shaughnessy, S.P. Hegarty, A.G. Vladimirov, H.C. Lyu, K. Karnowski, M. Wojtkowski, G. Huyet, Dynamics of a short cavity swept source OCT laser, Optics Express, 22 (2014) pp. 18177--18185.

  • J. Sieber, O. Omel'chenko, M. Wolfrum, Controlling unstable chaos: Stabilizing chimera states by feedback, Physical Review Letters, 112 (2014) pp. 054102/1--054102/5.
    Abstract
    We present a control scheme that is able to find and stabilize a chaotic saddle in a system with a large number of interacting particles. This allows us to track a high dimensional chaotic attractor through a bifurcation where it loses its attractivity. Similar to a classical delayed feedback control, the scheme is non-invasive, however, only in an appropriately relaxed sense considering the chaotic regime as a statistical equilibrium displaying random fluctuations as a finite size effects. We demonstrate the control scheme for so called chimera states, which are coherence-incoherence patterns in coupled oscillator systems. The control makes chimera states observable close to coherence, for small numbers of oscillators, and for random initial conditions.

  • O. Omel'chenko, M. Wolfrum, C. Laing, Partially coherent twisted states in arrays of coupled phase oscillators, Chaos. An Interdisciplinary Journal of Nonlinear Science, 24 (2014) pp. 023102/1--023102/9.
    Abstract
    We consider a one-dimensional array of phase oscillators with non-local coupling and a Lorentzian distribution of natural frequencies. The primary objects of interest are partially coherent states that are uniformly "twisted" in space. To analyze these we take the continuum limit, perform an Ott/Antonsen reduction, integrate over the natural frequencies and study the resulting spatio-temporal system on an unbounded domain. We show that these twisted states and their stability can be calculated explicitly. We find that stable twisted states with different wave numbers appear for increasing coupling strength in the well-known Eckhaus scenario. Simulations of finite arrays of oscillators show good agreement with results of the analysis of the infinite system.

  • R.M. Arkhipov, A. Pimenov, M. Radziunas, A.G. Vladimirov, D. Arsenjević, D. Rachinskii, H. Schmeckebier, D. Bimberg, Hybrid mode-locking in edge-emitting semiconductor lasers: Simulations, analysis and experiments, IEEE J. Select. Topics Quantum Electron., 19 (2013) pp. 1100208/1--1100208/6.
    Abstract
    Hybrid mode-locking in a two section edge-emitting semiconductor laser is studied numerically and analytically using a set of three delay differential equations. In this set the external RF signal applied to the saturable absorber section is modeled by modulation of the carrier relaxation rate in this section. Estimation of the locking range where the pulse repetition frequency is synchronized with the frequency of the external modulation is performed numerically and the effect of the modulation shape and amplitude on this range is investigated. Asymptotic analysis of the dependence of the locking range width on the laser parameters is carried out in the limit of small signal modulation. Our numerical simulations indicate that hybrid mode-locking can be also achieved in the cases when the frequency of the external modulation is approximately twice larger and twice smaller than the pulse repetition frequency of the free running passively mode-locked laser fP . Finally, we provide an experimental demonstration of hybrid mode-locking in a 20 GHz quantum-dot laser with the modulation frequency of the reverse bias applied to the absorber section close to fP =2.

  • M. Kantner, S. Yanchuk, Bifurcation analysis of delay-induced patterns in a ring of Hodgkin--Huxley neurons, Philosophical Transactions of the Royal Society A : Mathematical, Physical & Engineering Sciences, 371 (2013) pp. 20120470/1--20120470/13.

  • M.V. Arkhipov, R.M. Arkhipov, S.A. Pulkin, Effects of inversionless oscillation in two-level media from the point of view of specificities of the spatiotemporal propagation dynamics of radiation, Optics and Spectroscopy, 114 (2013) pp. 831--837.
    Abstract
    We report the results of computer simulation of the emission of radiation by an extended twolevel medium in a ring cavity. The cases of using strong external monochromatic, quasimonochromatic, and biharmonic radiation for pumping the twolevel medium are analyzed. It is shown that the emission of radiation with spectral content different from that of the pump radiation, which is interpreted as the inversionless oscillation, is the result of the spatiotemporal dynamics of light propagation in an extended twolevel medium imbedded in a cavity. The appearance of this radiation is not related to known resonances of amplification of a weak probe field in a thin layer of the twolevel system (the effect of inversionless oscillation) induced by strong resonance monochromatic or biharmonic field, as was thought before.

  • R. Čiegis, A. Mirinavičius, M. Radziunas, Comparison of split step solvers for multidimensional Schrödinger problems, Computational Methods in Applied Mathematics, 13 (2013) pp. 237--250.
    Abstract
    Hybrid mode-locking in a two section edge-emitting semiconductor laser is studied numerically and analytically using a set of three delay differential equations. In this set the external RF signal applied to the saturable absorber section is modeled by modulation of the carrier relaxation rate in this section. Estimation of the locking range where the pulse repetition frequency is synchronized with the frequency of the external modulation is performed numerically and the effect of the modulation shape and amplitude on this range is investigated. Asymptotic analysis of the dependence of the locking range width on the laser parameters is carried out in the limit of small signal modulation. Our numerical simulations indicate that hybrid mode-locking can be also achieved in the cases when the frequency of the external modulation is approximately twice larger and twice smaller than the pulse repetition frequency of the free running passively mode-locked laser fP . Finally, we provide an experimental demonstration of hybrid mode-locking in a 20 GHz quantum-dot laser with the modulation frequency of the reverse bias applied to the absorber section close to fP =2.

  • J. Sieber, M. Wolfrum, M. Lichtner, S. Yanchuk, On the stability of periodic orbits in delay equations with large delay, Discrete and Continuous Dynamical Systems, 33 (2013) pp. 3109--3134.
    Abstract
    We prove a necessary and sufficient criterion for the exponential stability of periodic solutions of delay differential equations with large delay. We show that for sufficiently large delay the Floquet spectrum near criticality is characterized by a set of curves, which we call asymptotic continuous spectrum, that is independent on the delay.

  • O. Omel'chenko, M. Wolfrum, Bifurcations in the Sakaguchi--Kuramoto model, Physica D. Nonlinear Phenomena, 263 (2013) pp. 74--85.
    Abstract
    We analyze the Sakaguchi-Kuramoto model of coupled phase oscillators in a continuum limit given by a frequency dependent version of the Ott-Antonsen system. Based on a self-consistency equation, we provide a detailed analysis of partially synchronized states, their bifurcation from the completely incoherent state and their stability properties. We use this method to analyze the bifurcations for various types of frequency distributions and explain the appearance of non-universal synchronization transitions.

  • O. Omel'chenko, Coherence-incoherence patterns in a ring of non-locally coupled phase oscillators, Nonlinearity, 26 (2013) pp. 2469--2498.
    Abstract
    We consider a paradigmatic spatially extended model of non-locally coupled phase oscillators which are uniformly distributed within a one-dimensional interval and interact depending on the distance between their sites modulo periodic boundary conditions. This model can display peculiar spatio-temporal patterns consisting of alternating patches with synchronized (coherent) or irregular (incoherent) oscillator dynamics, hence the name coherence-incoherence pattern, or chimera state. For such patterns we formulate a general bifurcation analysis scheme based on a hierarchy of continuum limit equations. This gives us possibility to classify known coherence-incoherence patterns and to suggest directions for searching new ones.

  • B. Fiedler, C. Rocha, M. Wolfrum, A permutation characterization of Sturm global attractors of Hamiltonian type, Journal of Differential Equations, 252 (2012) pp. 588--623.
    Abstract
    We consider Neumann boundary value problems of the form $u_t = u_xx + f $ on the interval $0 leq x leq pi$ for dissipative nonlinearities $f = f (u)$. A permutation characterization for the global attractors of the semiflows generated by these equations is well known, even in the general case $f = f (x, u, u_x )$. We present a permutation characterization for the global attractors in the restrictive class of nonlinearities $f = f (u)$. In this class the stationary solutions of the parabolic equation satisfy the second order ODE $v^primeprime + f (v) = 0$ and we obtain the permutation characterization from a characterization of the set of $2pi$-periodic orbits of this planar Hamiltonian system. Our results are based on a diligent discussion of this mere pendulum equation.

  • B. Fiedler, C. Rocha, M. Wolfrum, Sturm global attractors for S$^1$-equivariant parabolic equations, Networks Heterogeneous Media, 7 (2012) pp. 617--659.
    Abstract
    We consider Neumann boundary value problems of the form $u_t = u_xx + f $ on the interval $0 leq x leq pi$ for dissipative nonlinearities $f = f (u)$. A permutation characterization for the global attractors of the semiflows generated by these equations is well known, even in the general case $f = f (x, u, u_x )$. We present a permutation characterization for the global attractors in the restrictive class of nonlinearities $f = f (u)$. In this class the stationary solutions of the parabolic equation satisfy the second order ODE $v^primeprime + f (v) = 0$ and we obtain the permutation characterization from a characterization of the set of $2pi$-periodic orbits of this planar Hamiltonian system. Our results are based on a diligent discussion of this mere pendulum equation.

  • CH. Otto, K. Lüdge, A.G. Vladimirov, M. Wolfrum, E. Schöll, Delay induced dynamics and jitter reduction of passively mode-locked semiconductor lasers subject to optical feedback, New Journal of Physics, 14 (2012) pp. 113033/1-113033/29.

  • T. Girnyk, M. Hasler, Y. Maistrenko, Multistability of twisted states in non-locally coupled Kuramoto-type models, Chaos, Solitons and Fractals, 22 (2012) pp. 013114/1--013114/10.
    Abstract
    A ring of N identical phase oscillators with interactions between L-nearest neighbors is considered, where L ranges from 1 (local coupling) to N/2 (global coupling). The coupling function is a simple sinusoid, as in the Kuramoto model, but with a minus sign which has a profound influence on its behavior. Without limitation of the generality the frequency of the free-running oscillators can be set to zero. The resulting system is of gradient type and therefore all its solutions converge to an equilibrium point. All so-called q-twisted states, where the phase difference between neighboring oscillators on the ring is 2 pi q/N are equilibrium points, where q is an integer. Their stability in the limit N -> inf. is discussed along the line of1. In addition we prove that when a twisted state is asymptotically stable for the infinite system, it is also asymptotically stable for sufficiently large N. Note that for smaller N, the same q-twisted states may become unstable and other q-twisted states may become stable. Finally, the existence of additional equilibrium states, called here multi-twisted states, is shown by numerical simulation. The phase difference between neighboring oscillators is approximately 2 pi q/N in one sector of the ring, -2 pi q/N in another sector, and it has intermediate values between the two sectors. Our numerical investigation suggests that the number of different stable multi-twisted states grows exponentially as N -> inf. It is possible to interpret the equilibrium points of the coupled phase oscillator network as trajectories of a discrete-time translational dynamical system where the space-variable (position on the ring) plays the role of time. The q-twisted states are then fixed points and the multi-twisted states are periodic solutions of period N that are close to a heteroclinic cycle. Due to the apparently exponentially fast growing number of such stable periodic solutions, the system shows spatial chaos as N -> 1.

  • O. Omel'chenko, M. Wolfrum, Nonuniversal transitions to synchrony in the Sakaguchi--Kuramoto model, Physical Review Letters, 109 (2012) pp. 164101/1--164101/4.
    Abstract
    We investigate the transition to synchrony in a system of phase oscillators that are globally coupled with a phase lag (Sakaguchi-Kuramoto model). We show that for certain unimodal frequency distributions there appear unusual types of synchronization transitions, where synchrony can decay with increasing coupling, incoherence can regain stability for increasing coupling, or multistability between partially synchronized states and/or the incoherent state can appear. Our method is a bifurcation analysis based on a frequency dependent version of the Ott-Antonsen method and allows for a universal description of possible synchronization transition scenarios for any given distribution of natural frequencies.

  • O. Omel'chenko, M. Wolfrum, S. Yanchuk, Y. Maistrenko, O. Sudakov, Stationary patterns of coherence and incoherence in two-dimensional arrays of non-locally coupled phase oscillators, Phys. Rev. E (3), 85 (2012) pp. 036210/1--036210/5.
    Abstract
    Recently it has been shown that large arrays of identical oscillators with non-local coupling can have a remarkable type of solutions that display a stationary macroscopic pattern of coexisting regions with coherent and incoherent motion, often caled chimera states. We present here a detailed numerical study of the appearance of such solutions in two-dimensional arrays of coupled phase oscillators. We discover a variety of stationary patterns, including circular spots, stripe patterns, and patterns of multiple spirals. Here, the stationarity means that for increasing system size the locally averaged phase distributions tend to the stationary profile given by the corresponding thermodynamic limit equation.

  • M. Wolfrum, The Turing bifurcation in network systems: Collective patterns and single differentiated nodes, Physica D. Nonlinear Phenomena, 241 (2012) pp. 1351--1357.
    Abstract
    We study the emergence of patterns in a diffusively coupled network that undergoes a Turing instability. Our main focus is the emergence of stable solutions with a single differentiated node in systems with large and possibly irregular network topology. Based on a mean-field approach, we study the bifurcations of such solutions for varying system parameters and varying degree of the differentiated node. Such solutions appear typically before the onset of Turing instability and provide the basis for the complex scenario of multistability and hysteresis that can be observed in such systems. Moreover, we discuss the appearance of stable collective patterns and present a codimension-two bifurcation that organizes the interplay between collective patterns and patterns with single differentiated nodes.

  • A.G. Vladimirov, R. Lefever, M. Tlidi, Relative stability of multipeak localized patterns of cavity solitons, Physical Review A, 84 (2011) pp. 043848/1--043848/4.
    Abstract
    We study the relative stability of different one-dimensional (1D) and two-dimensional (2D) clusters of closely packed localized peaks of the Swift-Hohenberg equation. In the 1D case, we demonstrate numerically the existence of a spatial Maxwell transition point where all clusters involving up to 15 peaks are equally stable. Above (below) this point, clusters become more (less) stable when their number of peaks increases. In the 2D case, since clusters involving more than two peaks may exhibit distinct spatial arrangements, this point splits into a set of Maxwell transition pointsWe study the relative stability of different one-dimensional (1D) and two-dimensional (2D) clusters of closely packed localized peaks of the Swift-Hohenberg equation. In the 1D case, we demonstrate numerically the existence of a spatial Maxwell transition point where all clusters involving up to 15 peaks are equally stable. Above (below) this point, clusters become more (less) stable when their number of peaks increases. In the 2D case, since clusters involving more than two peaks may exhibit distinct spatial arrangements, this point splits into a set of Maxwell transition points

  • I. Babushkin, U. Bandelow, A. Vladimirov, Rotational symmetry breaking in small-area circular vertical cavity surface emitting lasers, Optics Communications, 284 (2011) pp. 1299--1302.
    Abstract
    We investigate theoretically the dynamics of three low-order transverse modes in a small-area vertical cavity surface emitting laser. We demonstrate the breaking of axial symmetry of the transverse field distribution in such a device. In particular, we show that if the linewidth enhancement factor is sufficiently large dynamical regimes with broken axial symmetry can exist up to very high diffusion coefficients  10 um^2/ns.

  • M. Lichtner, M. Wolfrum, S. Yanchuk, The spectrum of delay differential equations with large delay, SIAM Journal on Mathematical Analysis, 43 (2011) pp. 788-802.

  • M. Wolfrum, O. Omel'chenko, Chimera states are chaotic transients, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 84 (2011) pp. 015201(R)/1--015201(R)/4.
    Abstract
    Spatiotemporal chaos and turbulence are universal concepts for the explanation of irregular behavior in various physical systems. Recently, a remarkable new phenomenon, called "chimera states", has been described, where in a spatially homogeneous system regions of irregular incoherent motion coexist with regular synchronized motion, forming a self organized pattern in a population of nonlocally coupled oscillators. Whereas most of the previous studies of chimera states focused their attention to the case of large numbers of oscillators employing the thermodynamic limit of infinitely many oscillators, we investigate here the properties of chimera states in populations of finite size using concepts from deterministic chaos. Our calculations of the Lyapunov spectrum show that the incoherent motion, which is described in the thermodynamic limit as a stationary behavior, in finite size systems appears as weak spatially extensive chaos. Moreover, for sufficiently small populations the chimera states reveal their transient nature: after a certain time-span we observe a sudden collapse of the chimera pattern and a transition to the completely coherent state. Our results indicate that chimera states can be considered as chaotic transients, showing the same properties as type-II supertransients in coupled map lattices.

  • M. Wolfrum, O. Omel'chenko, S. Yanchuk, Y. Maistrenko, Spectral properties of chimera states, Chaos. An Interdisciplinary Journal of Nonlinear Science, 21 (2011) pp. 0013112/1-013112/8.
    Abstract
    Chimera states are particular trajectories in systems of phase oscillators with nonlocal coupling that display a pattern of coherent and incoherent motion. We present here a detailed analysis of the spectral properties for such trajectories. First, we study numerically their Lyapunov spectrum and its behavior for an increasing number of oscillators. The spectra demonstrate the hyperchaotic nature of the chimera states and show a correspondence of the Lyapunov dimension with the number of incoherent oscillators. Then, we pass to the thermodynamic limit equation and present an analytic approach to the spectrum of a corresponding linearized evolution operator. We show that in this setting, the chimera state is neutrally stable and that the continuous spectrum coincides with the limit of the hyperchaotic Lyapunov spectrum obtained for the finite size systems.

  • A.G. Vladimirov, U. Bandelow, G. Fiol, D. Arsenijević, M. Kleinert, D. Bimberg, A. Pimenov, D. Rachinskii, Dynamical regimes in a monolithic passively mode-locked quantum dot laser, Journal of the Optical Society of America. B, 27 (2010) pp. 2102-2109.

  • P. Perlikowski, S. Yanchuk, M. Wolfrum, A. Stefanski, P. Mosiolek, T. Kapitaniak, Routes to complex dynamics in a ring of unidirectionally coupled systems, Chaos. An Interdisciplinary Journal of Nonlinear Science, 20 (2010) pp. 013111/1--013111/9.

  • M. Tlidi, A.G. Vladimirov, D. Turaev, G. Kozyreff, D. Pieroux, T. Erneux, Spontaneous motion of localized structures and localized patterns induced by delayed feedback, The European Physical Journal D. Atomic, Molecular, Optical and Plasma Physics, 59 (2010) pp. 59-65.

  • O.E. Omel'chenko, Y.L. Maistrenko, P.A. Tass, Chimera states induced by spatially modulated delayed feedback, Phys. Rev. E (3), 82 (2010) pp. 066201/1--066201/13.

  • O.E. Omel'chenko, M. Wolfrum, Y.L. Maistrenko, Chimera states as chaotic spatio-temporal patterns, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 81 (2010) pp. 065201(R)/1--065201R)/4.

  • M. Wolfrum, S. Yanchuk, A multiple time scale approach to the stability of external cavity modes in the Lang--Kobayashi system using the limit of large delay, SIAM Journal on Applied Dynamical Systems, 9 (2010) pp. 519--535.

  • M. Wolfrum, S. Yanchuk, P. Hövel, E. Schöll, Complex dynamics in delay-differential equations with large delay, The European Physical Journal, Special Topics, 191 (2010) pp. 91--103.

  • M. Tlidi, A.G. Vladimirov, D. Pieroux, D. Turaev, Spontaneous motion of cavity solitons induced by a delayed feedback, Physical Review Letters, 103 (2009) pp. 103904/1--103904/4.

  • V.Z. Tronciu, Excitability and coherence resonance of DFB laser with passive dispersive reflector, Moldavian Journal of the Physical Sciences, 7 (2008) pp. 218-223.

  • S. Yanchuk, M. Wolfrum, Destabilization patterns in large regular networks, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 77 (2008) pp. 026212/1-026212/7.
    Abstract
    We describe a generic mechanism for the destabilization in large regular networks of identical coupled oscillators. Based on a reduction method for the spectral problem, we first present a criterion for this type of destabilization. Then, we investigate the related bifurcation scenario, showing the existence of a large number of coexisting periodic solutions with different frequencies, spatial patterns, and stability properties. Even for unidirectional coupling this can be understood in analogy to the well-known Eckhaus scenario for diffusive systems.

  • V.F. Butuzov, N.N. Nefedov, L. Recke, K.R. Schneider, Existence and stability of solutions with periodically moving weak internal layers, Journal of Mathematical Analysis and Applications, 348 (2008) pp. 508-515.
    Abstract
    We consider the periodic parabolic differential equation $ep^2 Big( fracpartial^2 upartial x^2 -fracpartial upartial t Big)=f(u,x,t,ep)$ under the assumption that $ve$ is a small positive parameter and that the degenerate equation $f(u,x,t,0) =0$ has two intersecting solutions. We derive conditions such that there exists an asymptotically stable solution $u_p(x,t,ep)$ which is $T$-periodic in $t$, satisfies no-flux boundary conditions and tends to the stable composed root of the degenerate equation as $eprightarrow 0$.

  • D. Turaev, M. Radziunas, A.G. Vladimirov, Chaotic soliton walk in periodically modulated media, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 77 (2008) pp. 06520/1--06520/4.

  • M. Lichtner, Spectral mapping theorem for linear hyperbolic systems, Proceedings of the American Mathematical Society, 136 (2008) pp. 2091-2101.

  • D. Turaev, A.G. Vladimirov, S. Zelik, Chaotic bound state of localized structures in the complex Ginzburg--Landau equation, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 75 (2007) pp. 045601/1-045601/4.

  • M. Lichtner, M. Radziunas, L. Recke, Well-posedness, smooth dependence and center manifold reduction for a semilinear hyperbolic system from laser dynamics, Mathematical Methods in the Applied Sciences, 30 (2007) pp. 931--960.

  • A.G. Vladimirov, D.V. Skryabin, G. Kozyreff, P. Mandel, M. Tlidi, Bragg localized structures in a passive cavity with transverse modulation of the refractive index and the pump, Optics Express, 14 (2006) pp. 1--6.

  • S. Yanchuk, A. Stefanski, T. Kapitaniak, J. Wojewoda, Dynamics of an array of mutually coupled semiconductor lasers, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 73 (2006) pp. 016209/1--016209/7.

  • S. Yanchuk, M. Wolfrum, P. Hövel, E. Schöll, Control of unstable steady states by strongly delayed feedback, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 74 (2006) pp. 026201/1--026201/7.

  • M. Nizette, D. Rachinskii, A. Vladimirov, M. Wolfrum, Pulse interaction via gain and loss dynamics in passive mode-locking, Physica D. Nonlinear Phenomena, 218 (2006) pp. 95--104.

  • A. Politi, F. Ginelli, S. Yanchuk, Y. Maistrenko, From synchronization to Lyapunov exponents and back, Physica D. Nonlinear Phenomena, 224 (2006) pp. 90-101.

  • D.I. Rachinskii, A. Vladimirov, U. Bandelow, B. Hüttl, R. Kaiser, Q-switching instability in a mode-locked semiconductor laser, Journal of the Optical Society of America. B, 23 (2006) pp. 663--670.

  • A. Yulin, D. Skryabin, A.G. Vladimirov, Modulation instability of discrete solitons in coupled waveguides with group velocity dispersion, Optics Express, 14 (2006) pp. 12347--12352.

  • M. Wolfrum, S. Yanchuk, Eckhaus instability in systems with large delay, Physical Review Letters, 96 (2006) pp. 220201/1--220201/4.

  • TH. Koprucki, M. Baro, U. Bandelow, Th. Tien, F. Weik, J.W. Tomm, M. Grau, M.-Ch. Amann, Electronic structure and optoelectronic properties of strained InAsSb/GaSb multiple quantum wells, Applied Physics Letters, 87 (2005) pp. 181911/1--181911/3.

  • A. Vladimirov, D. Turaev, Model for passive mode locking in semiconductor lasers, Physical Review A, 72 (2005) pp. 033808/1-033808/13.

  • S. Yanchuk, Discretization of frequencies in delay coupled oscillators, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 72 (2005) pp. 036205/1-036205/5.

  • S. Yanchuk, Properties of stationary states of delay equations with large delay and applications to laser dynamics, Mathematical Methods in the Applied Sciences, 28 (2005) pp. 363--377.

  • K. Gatermann, M. Wolfrum, Bernstein's second theorem and Viro's method for sparse polynomial systems in chemistry, Adv. Appl. Math., 34 (2005) pp. 252--294.

  • D.I. Rachinskii, K.R. Schneider, Dynamic Hopf bifurcations generated by nonlinear terms, Journal of Differential Equations, 210 (2005) pp. 65--86.

  • M. Wolfrum, J. Härterich, Describing a class of global attractors via symbol sequences, Discrete and Continuous Dynamical Systems, 12 (2005) pp. 531-554.

  • S. Yanchuk, A. Stefanski, J. Wojewoda, T. Kapitaniak, Simple estimation of synchronization threshold in ensembles of diffusively coupled chaotic systems, Phys. Rev. E (3), 70 (2004) pp. 026217/1--026217/11.

  • B. Fiedler, C. Rocha, M. Wolfrum, Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle, Journal of Differential Equations, 201 (2004) pp. 99--138.

  • K.R. Schneider, S. Yanchuk, L. Recke, Dynamics of two mutually coupled semiconductor lasers: Instantaneous coupling limit, Phys. Rev. E (3), 69 (2004) pp. 056221/1--056221/12.

  • S.V. Fedorov, N.N. Rosanov, A.N. Shatsev, N.A. Veretenov, A.G. Vladimirov, Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with saturable absorber, IEEE J. Quantum Electron., 39 (2003) pp. 216--226.

  • D. Rachinskii, K.R. Schneider, Delayed loss of stability in systems with degenerate linear parts, Zeitschrift fur Analysis und ihre Anwendungen. Journal for Analysis and its Applications, 22 (2003) pp. 433--453.

  • M. Tlidi, A.G. Vladimirov, P. Mandel, Interaction and stability of periodic and localized structures in optical bistable systems, IEEE J. Quantum Electron., 39 (2003) pp. 197--205.

  • K.R. Schneider, E. Shchetinina, One-parametric families of canard cycles: Two explicitly solvable examples, Mathematical Methods in the Applied Sciences, 2 (2003) pp. 74-75.

  • K.R. Schneider, S. Yanchuk, Complete synchronization of symmetrically coupled autonomous systems, Applicable Analysis. An International Journal, 82 (2003) pp. 1127-1143.

  • K.R. Schneider, V.A. Sobolev, E. Shchepakina, New type of travelling wave solutions, Mathematical Methods in the Applied Sciences, 26 (2003) pp. 1349-1361.

  • A.G. Vladimirov, G. Kozyreff, P. Mandel, Synchronization of weakly stable oscillators and semiconductor laser arrays, Europhysics Letters, 61 (2003) pp. 613--619.

  • A.G. Vladimirov, G. Kozyreff, P. Mandel, Synchronization of weakly stable oscillators and semiconductor laser arrays, Europhysics Letters, 61 (2003) pp. 613--619.

  • S. Yanchuk, T. Kapitaniak, Manifestation of riddling in the presence of small parameter mismatch between coupled systems, Phys. Rev. E (3), 68, 017202 (2003) pp. 4.

  • S. Yanchuk, Y. Maistrenko, E. Mosekilde, Synchronization of time-continuous chaotic oscillators, Chaos. An Interdisciplinary Journal of Nonlinear Science, 13 (2003) pp. 388--400.

  • S. Yanchuk, G. Kristensen, I. Shushko, Dynamical approach to complex regional economic growth based on Keynesian model for China, Chaos, Solitons and Fractals, 18 (2003) pp. 937--952.

  • M. Wolfrum, A sequence of order relations: Encoding heteroclinic connections in scalar parabolic PDE, Journal of Differential Equations, 183 (2002) pp. 56--78.

  • M. Wolfrum, Geometry of heteroclinic cascades in scalar parabolic differential equations, Journal of Dynamics and Differential Equations, 14 (2002) pp. 207--241.

  Contributions to Collected Editions

  • W.W. Ahmed, S. Kumar, R. Herrero, M. Botey, M. Radziunas, K. Staliunas, Suppression of modulation instability in pump modulated flat-mirror VECSELs, in: Nonlinear Optics and its Applications IV, B.J. Eggleton, N.G.R. Broderick, A.L. Gaeta, eds., 9894 of Proceedings of SPIE, SPIE Digital Library, 2016, pp. 989406/1--989406/7.
    Abstract
    We show that modulation instability (MI) can be suppressed in vertical external cavity surface emitting lasers (VECSELs) by introducing a periodic spatio-temporal modulation of the pump profile which in turn allows a simple flat-mirror configuration. The stability analysis of such pump modulated flat-mirror VECSELs is performed by a modified Floquet method and results are confirmed by full numerical integration of the model equations. It is found that the amplitude of the modulation as well as its spatial and temporal frequencies are crucial parameters for high spatial beam quality emission. We identify regions of complete and partial stabilization in parameter space for VECSELs with different external cavity lengths. The proposed method is shown to efficiently stabilize VECSELs with cavity lengths ranging from millimetres up to centimetres. However, the applicability of this method becomes limited for micro-meter-long cavities due to strong intrinsic relaxation oscillations.

  • D. Turaev, A.G. Vladimirov, S. Zelik, Interaction of spatial and temporal cavity solitons in mode-locked lasers and passive cavities, in: Laser Optics (LO), 2016 International Conference, IEEE, New York, 2016, pp. 37628.
    Abstract
    We study interaction of well-separated localized structures of light in the presence of periodic perturbations. Oscillating localized structures were found to emit weakly decaying dispersive waves leading to a strong enhancement of the interaction and formation of new types of bound states. We discuss the applicability of our analytical results to the interpretation of experimental and numerical data reported earlier.

  • A.G. Vladimirov, G. Huyet, A. Pimenov, Delay differential models in multimode laser dynamics: Taking chromatic dispersion into account, in: Semiconductor Lasers and Laser Dynamics VII, 9892 of Proceedings of SPIE, SPIE, Bellingham, Washington, 2016, pp. 98920I/1--98920I/7.
    Abstract
    A set of differential equations with distributed delay is derived for modeling of multimode ring lasers with intracavity chromatic dispersion. Analytical stability analysis of continuous wave regimes is performed and it is demonstrated that sufficiently strong anomalous dispersion can destabilize these regimes. © (2016) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.

  • U. Bandelow, S. Amiranashvili, N. Akhmediev, Limitation for ultrashort solitons in nonlinear optical fibers by cusp formation, in: CLEO®/Europe -- EQEC 2015: Conference Digest, OSA Technical Digest (Online) (Optical Society of America, 2015), paper EI-2.3 THU, 2015, pp. 1--1.

  • A. Glitzky, A. Mielke, L. Recke, M. Wolfrum, S. Yanchuk, D2 -- Mathematics for optoelectronic devices, in: MATHEON -- Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 243--256.

  • S. Amiranashvili, A. Demircan, C. Brée, G. Steinmeyer, F. Mitschke, Manipulating light by light in optical fibers, in: 3rd Bonn Humboldt Award Winners' Forum ``Frontiers in Quantum Optics: Taming the World of Atoms and Photons -- 100 Years after Niels Bohr'', Bonn, October 9--12, 2013, Networking Guide, pp. 58--59.

  • P. Perlikowski, S. Yanchuk, M. Wolfrum, A. Stefanski, T. Kapitaniak, Dynamics of a large ring of unidirectionally coupled duffing oscillators, in: IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design, Proceedings of the IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design, held Aberdeen, UK, 27--30 July 2010, M. Wiercigroch, G. Rega, eds., 32 of IUTAM Bookseries, Springer, Dordrecht et al., 2013, pp. 63--72.

  • C. Brée, S. Amiranashvili, U. Bandelow, Spatio-temporal pulse propagation in nonlinear dispersive optical media, in: Proceedings of the 12th International Conference on Numerical Simulation of Optoelectronic Devices, NUSOD'12, J. Piprek, W. Lu, eds., IEEE Conference Publications Management Group, New Jersey, USA, 2012, pp. 131--132.

  • D. Turaev, A.G. Vladimirov, S. Zelik, Strong enhancement of interaction of optical pulses induced by oscillatory instability, in: CLEO/Europe and EQEC 2009 Conference Digest (Optical Society of America, 2009), poster EH.P.13 WED, 2009, pp. 1--1.

  • L. Recke, M. Wolfrum, S. Yanchuk, Dynamics of coupled semiconductor lasers, in: Analysis and Control of Complex Nonlinear Processes in Physics, Chemistry and Biology, Chapter 6, L. Schimansky-Geier, B. Fiedler, J. Kurths, E. Schöll, eds., World Scientific, New Jersey [et al.], 2007, pp. 185--212.

  • A.G. Vladimirov, D.V. Skryabin, M. Tlidi, Localized structures of light in nonlinear devices with intracavity photonic bandgap material, in: 2007 European Conference on Lasers and Electro-Optics and the European Quantum Electronics Conference (CLEO®/Europe-IQEC) Conference Digest (oral presentation IG-4-MON), IEEE, 2007, pp. 1--1.

  • J. Ehrt, J. Härterich, Convergence to stationary states in spatially inhomogeneous balance laws, in: Hyperbolic Problems. Theory, Numerics and Applications -I-, F. Asakura, S. Kawashima, A. Matsumura, S. Nishibata, K. Nishihara, eds., Yokohama Publishers, Yokohama, 2006, pp. 367--374.

  • M. Wolfrum, The concept of adjacency for stationary and non-stationary solutions of scalar semilinear parabolic PDE, in: EQUADIFF 2003. Proceedings of the International Conference on Differential Equations, Hasselt, Belgium, 22--26 July 2003, F. Dumortier, H. Broer, J. Mawhin, A. Vanderbauwhede, S. Verduyn Lunel, eds., World Scientific, Singapore, 2005, pp. 678--684.

  • S. Yanchuk, K.R. Schneider, Complete synchronization of symmetrically coupled autonomous systems, in: EQUADIFF 2003. Proceedings of the International Conference on Differential Equations, Hasselt, Belgium, 22--26 July 2003, F. Dumortier, H. Broer, J. Mawhin, A. Vanderbauwhede, S. Verduyn Lunel, eds., World Scientific, Singapore, 2005, pp. 494--496.

  • S. Yanchuk, M. Wolfrum, Instabilities of equilibria of delay-differential equations with large delay, in: Proceedings of the ENOC-2005 Fifth EUROMECH Nonlinear Dynamics Conference, Eindhoven, The Netherlands, August 7--12, 2005, D.H. VAN Campen, M.D. Lazurko, W.P.J.M. caps">caps">van der Oever, eds., 2005, pp. 1060--1065.

  • M. Wolfrum, S. Yanchuk, Synchronous and asynchronous instabilities of two lasers with a long delayed coupling, in: Proceedings of the ENOC-2005 Fifth EUROMECH Nonlinear Dynamics Conference, Eindhoven, The Netherlands, August 7--12, 2005, D.H. VAN Campen, M.D. Lazurko, W.P.J.M. caps">caps">van der Oever, eds., 2005, pp. 2069--2073.

  • S. Yanchuk, K.R. Schneider, L. Recke, Dynamics of two F2F coupled lasers: Instantaneous coupling limit, in: Proceeding of SPIE: Semiconductor Lasers and Laser Dynamics Conference ``Photonics Europe'', 5452, SPIE, Washington, USA, 2004, pp. 51--62.

  • Y.L. Maistrenko, O. Popovych, S. Yanchuk, Synchronization and clustering in ensembles of coupled chaotic oscillators, in: Synchronization: Theory and Application. Proceedings of the NATO Advanced Study Institute, A. Pikovsky, Y.L. Maistrenko, eds., 109 of NATO Sci. Ser. II Math. Phys. Chem., Kluwer Acad. Publishers, Dordrecht, 2003, pp. 101--138.

  Preprints, Reports, Technical Reports

  • A.P. Willis, Y. Duguet, O.E. Omel'chenko, M. Wolfrum, Surfing the edge: Finding nonlinear solutions using feedback control, Preprint no. 2389, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2389 .
    Abstract, PDF (587 kByte)
    Many transitional wall-bounded shear flows are characterised by the coexistence in state-space of laminar and turbulent regimes. Probing the edge boundarz between the two attractors has led in the last decade to the numerical discovery of new (unstable) solutions to the incompressible Navier--Stokes equations. However, the iterative bisection method used to achieve this can become prohibitively costly for large systems. Here we suggest a simple feedback control strategy to stabilise edge states, hence accelerating their numerical identification by several orders of magnitude. The method is illustrated for several configurations of cylindrical pipe flow. Traveling waves solutions are identified as edge states, and can be isolated rapidly in only one short numerical run. A new branch of solutions is also identified. When the edge state is a periodic orbit or chaotic state, the feedback control does not converge precisely to solutions of the uncontrolled system, but nevertheless brings the dynamics very close to the original edge manifold in a single run. We discuss the opportunities offered by the speed and simplicity of this new method to probe the structure of both state space and parameter space.

  • S. Olmi, D. Angulo-Garcia, A. Imparato, A. Torcini, Exact firing time statistics of neurons driven by discrete inhibitory noise, Preprint no. 2367, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2367 .
    Abstract, PDF (364 kByte)
    Neurons in the intact brain receive a continuous and irregular synaptic bombardment from excitatory and inhibitory pre-synaptic neurons, which determines the firing activity of the stimulated neuron. In order to investigate the influence of inhibitory stimulation on the firing time statistics, we consider Leaky Integrate-and-Fire neurons subject to inhibitory instantaneous post-synaptic potentials. In particular, we report exact results for the firing rate, the coefficient of variation and the spike train spectrum for various synaptic weight distributions. Our results are not limited to stimulations of infinitesimal amplitude, but they apply as well to finite amplitude post-synaptic potentials, thus being able to capture the effect of rare and large spikes. The developed methods are able to reproduce also the average firing properties of heterogeneous neuronal populations.

  • K. Disser, Global existence, uniqueness and stability for nonlinear dissipative systems of bulk-interface interaction, Preprint no. 2313, WIAS, Berlin, 2016.
    Abstract, PDF (302 kByte)
    We consider a general class of nonlinear parabolic systems corresponding to thermodynamically consistent gradient structure models of bulk-interface interaction. The setting includes non-smooth geometries and e.g. slow, fast and "entropic'' diffusion processes under mass conservation. The main results are global well-posedness and exponential stability of equilibria. As a part of the proof, we show bulk-interface maximum principles and a bulk-interface Poincaré inequality. The method of proof for global existence is a simple but very versatile combination of maximal parabolic regularity of the linearization, a priori L-bounds and a Schaefer's fixed point argument. This allows us to extend the setting e.g. to Allen-Cahn dissipative dynamics and to include large classes of inhomogeneous boundary conditions and external forces.

  • O. Omel'chenko, L. Recke, V. Butuzov, N. Nefedov, Time-periodic boundary layer solutions to singularly perturbed parabolic problems, Preprint no. 2300, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2300 .
    Abstract, PDF (269 kByte)
    In this paper, we present a result of implicit function theorem type, which was designed for application to singularly perturbed problems. This result is based on fixed point iterations for contractive mappings, in particular, no monotonicity or sign preservation properties are needed. Then we apply our abstract result to time-periodic boundary layer solutions (which are allowed to be non-monotone with respect to the space variable) in semilinear parabolic problems with two independent singular perturbation parameters. We prove existence and local uniqueness of those solutions, and estimate their distance to certain approximate solutions.

  • M. Hintermüller, C.N. Rautenberg, M. Mohammadi, M. Kanitsar, Optimal sensor placement: A robust approach, Preprint no. 2287, WIAS, Berlin, 2016.
    Abstract, PDF (4835 kByte)
    We address the problem of optimally placing sensor networks for convection-diffusion processes where the convective part is perturbed. The problem is formulated as an optimal control problem where the integral Riccati equation is a constraint and the design variables are sensor locations. The objective functional involves a term associated to the trace of the solution to the Riccati equation and a term given by a constrained optimization problem for the directional derivative of the previous quantity over a set of admissible perturbations. The paper addresses the existence of the derivative with respect to the convective part of the solution to the Riccati equation, the well-posedness of the optimization problem and finalizes with a range of numerical tests.

  • K.R. Schneider, A. Grin, Construction of generalized pendulum equations with prescribed maximum number of limit cycles of the second kind, Preprint no. 2272, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2272 .
    Abstract, PDF (229 kByte)
    Consider a class of planar autonomous differential systems with cylindric phase space which represent generalized pendulum equations. We describe a method to construct such systems with prescribed maximum number of limit cycles which are not contractible to a point (limit cycles of the second kind). The underlying idea consists in employing Dulac-Cherkas functions. We also show how this approach can be used to control the bifurcation of multiple limit cycles.

  • K.R. Schneider, A. Grin, Study of the bifurcation of a multiple limit cycle of the second kind by means of a Dulac--Cherkas function: A case study, Preprint no. 2226, WIAS, Berlin, 2016.
    Abstract, PDF (174 kByte)
    We consider a generalized pendulum equation depending on the scalar parameter $mu$ having for $mu=0$ a limit cycle $Gamma$ of the second kind and of multiplicity three. We study the bifurcation behavior of $Gamma$ for $-1 le mu le (sqrt5+3)/2$ by means of a Dulac-Cherkas function.

  • B. Jahnel, Ch. Külske, Attractor properties for irreversible and reversible interacting particle systems, Preprint no. 2145, WIAS, Berlin, 2015.
    Abstract, PDF (262 kByte)
    We consider translation-invariant interacting particle systems on the lattice with finite local state space admitting at least one Gibbs measure as a time-stationary measure. The dynamics can be irreversible but should satisfy some mild non-degeneracy conditions. We prove that weak limit points of any trajectory of translation-invariant measures, satisfying a non-nullness condition, are Gibbs states for the same specification as the time-stationary measure. This is done under the additional assumption that zero entropy loss of the limiting measure w.r.t. the time-stationary measure implies that they are Gibbs measures for the same specification.We also give an alternate version of the last condition such that the non-nullness requirement can be dropped. For dynamics admitting a reversible Gibbs measure the alternative condition can be verified, which yields the attractor property for such dynamics. This generalizes convergence results using relative entropy techniques to a large class of dynamics including irreversible and non-ergodic ones. We use this to show synchronization for the rotation dynamics exhibited in citeJaKu12 possibly at low temperature, and possibly non-reversible. We assume the additional regularity properties on the dynamics: 1 There is at least one stationary measure which is a Gibbs measure. 2 Zero loss of relative entropy density under dynamics implies the Gibbs property.

  • V.F. Butuzov, N.N. Nefedov, L. Recke, K.R. Schneider, Asymptotics and stability of a periodic solution to a singularly perturbed parabolic problem in case of a double root of the degenerate equation, Preprint no. 2141, WIAS, Berlin, 2015.
    Abstract, PDF (162 kByte)
    For a singularly perturbed parabolic problem with Dirichlet conditions we prove the existence of a solution periodic in time and with boundary layers at both ends of the space interval in the case that the degenerate equation has a double root. We construct the corresponding asymptotic expansion in the small parameter. It turns out that the algorithm of the construction of the boundary layer functions and the behavior of the solution in the boundary layers essentially differ from that ones in case of a simple root. We also investigate the stability of this solution and the corresponding region of attraction.

  • L. Lücken, S. Yanchuk, Detection and storage of multivariate temporal sequences by spiking pattern reverberators, Preprint no. 2122, WIAS, Berlin, 2015.
    Abstract, PDF (876 kByte)
    We consider networks of spiking coincidence detectors in continuous time. A single detector is a finite state machine that emits a pulsatile signal whenever the number incoming inputs exceeds a threshold within a time window of some tolerance width. Such finite state models are well-suited for hardware implementations of neural networks, as on integrated circuits (IC) or field programmable arrays (FPGAs) but they also reflect the natural capability of many neurons to act as coincidence detectors. We pay special attention to a recurrent coupling structure, where the delays are tuned to a specific pattern. Applying this pattern as an external input leads to a self-sustained reverberation of the encoded pattern if the tuning is chosen correctly. In terms of the coupling structure, the tolerance and the refractory time of the individual coincidence detectors, we determine conditions for the uniqueness of the sustained activity, i.e., for the funcionality of the network as an unambiguous pattern detector. We also present numerical experiments, where the functionality of the proposed pattern detector is demonstrated replacing the simplistic finite state models by more realistic Hodgkin-Huxley neurons and we consider the possibility of implementing several pattern detectors using a set of shared coincidence detectors. We propose that inhibitory connections may aid to increase the precision of the pattern discrimination.

  Talks, Poster

  • S. Reichelt, Pulses in FitzHugh-Nagumo systems with periodic coefficients, Seminar ``Dynamical Systems and Applications'', Technische Universität Berlin, Institut für Mathematik, Berlin, May 3, 2017.

  • M. Kantner, Modeling of quantum dot based single-photon LEDs on a device level, MATHEON Workshop 10th Annual Meeting ``Photonic Devices'', February 9 - 10, 2017, Konrad-Zuse-Zentrum für Informationstechnik Berlin, February 10, 2017.

  • O. Omel'chenko, Bifurcations mediating the appearance of chimera states, SIAM Conference on Applications of Dynamical Systems (DS 17), Minisymposium "Large Scale Dynamics In Coupled Systems On Networks", May 21 - 25, 2017, Society for Industrial and Applied Mathematics (SIAM), Snowbird, USA, May 24, 2017.

  • O. Omel'chenko, Introduction to chimera states, Seminar of the Scientific Computing Laboratory, Institue of Physics, University of Belgrade, Serbia, May 4, 2017.

  • O. Omel'chenko, Stabilizing control scheme: From chimera states to edge states, Internal seminar of the Prof. E. Knobloch group, Department of Physics, University of California, Berkeley, USA, May 17, 2017.

  • M. Wolfrum, Chimera states in systems of coupled phase oscillators, Emerging Topics in Network Dynamical Systems, June 6 - 9, 2017, Lorentz Center, Leiden, Netherlands, June 6, 2017.

  • S. Reichelt, Competing patterns in anti-symmetrically coupled Swift--Hohenberg equations, International Conference on Control of Complex Systems and Networks, SFB 910 ``Control of Self-Organizing Nonlinear Systems: Theoretical Methods and Concepts of Application'', Heringsdorf/Usedom, September 4 - 8, 2016.

  • S. Eydam, Mode-locking in systems of coupled phase oscillators, Seminar Applied Dynamical Systems, Technische Universität Berlin, Berlin, July 13, 2016.

  • S. Eydam, Mode-locking in systems of phase oscillators with higher harmonic coupling, International Conference on Control of Complex Systems and Networks, SFB 910 ``Control of Self-Organizing Nonlinear Systems: Theoretical Methods and Concepts of Application'', Heringsdorf/Usedom, September 4 - 8, 2016.

  • S. Eydam, Mode-locking in systems of phase oscillators with higher harmonic interaction, Workshop on Synchronization and Oscillators with Generalized Coupling, Exeter, UK, April 20 - 22, 2016.

  • M. Kantner, Modeling and simulation of carrier dynamics in quantum dot based single-photon sources, Nonlinear Dynamics in Semiconductor Lasers, WIAS, Berlin, June 15, 2016.

  • U. Bandelow, Nonlinear dynamical effects in photonics: Modeling, simulation and analysis, Coloquio del Instituto de Física, Pontificia Universidad Católica de Chile, Santiago, December 14, 2016.

  • U. Bandelow, Ultrashort solitons that do not want to be too short in duration, XIX Conference on Nonequilibrium Statistical Mechanics and Nonlinear Physics (MEDYFINOL 2016), Universidad de los Andes, Universidad de Mar del Plata, and Instituto Tecnológico de Buenos Aires, Valdivia, Chile, December 7, 2016.

  • O. Omel'chenko, Asymptotics of traveling coherence-incoherence patterns, Contemporary Problems of Mathematical Physics and Computational Mathematics, Lomonosov Moscow State University, Russian Federation, November 2, 2016.

  • O. Omel'chenko, Chimera states in nonlocally coupled oscillators: Their variety and control, 4th International Conference on Complex Dynamical Systems and Applications, National Institute of Technology, Durgapur, India, February 16, 2016.

  • O. Omel'chenko, Creative control for chimera states, Workshop on Synchronization and Oscillators with Generalized Coupling, University of Exeter, UK, April 21, 2016.

  • O. Omel'chenko, Mean-field equation for coherence-incoherence patterns, 7th European Congress of Mathematics (7ECM), Minisymposium 37 ``Propagation Phenomena in Discrete Media'', July 18 - 22, 2016, Technische Universität Berlin, July 22, 2016.

  • O. Omel'chenko, Nonuniversal transitions to synchrony in globally coupled phase oscillator, International Workshop on Nonlinear Complex Dynamical Systems, Indian Statistical Institute, Kolkata, February 19, 2016.

  • O. Omel'chenko, On the limitations of the Kuramoto model, Dynamics Days Latin America and the Caribbean, Benemérita Universidad Autónoma de Puebla, Mexico, October 28, 2016.

  • O. Omel'chenko, Patterns of coherence and incoherence, Patterns of Dynamics Conference in Honor of Bernold Fiedler, July 25 - 29, 2016, Free University of Berlin, Berlin, July 29, 2016.

  • O. Omel'chenko, Regular and irregular patterns of self-localized excitation in arrays of coupled phase oscillators, International Conference on Control of Complex Systems and Networks, SFB 910 ``Control of Self-Organizing Nonlinear Systems: Theoretical Methods and Concepts of Application'', Heringsdorf/Usedom, September 4 - 8, 2016.

  • O. Omel'chenko, Spike solutions to singularly perturbed elliptic problems, The 13th Annual Workshop on Numerical Methods for Problems with Layer Phenomena, Lomonosov Moscow State University, Russian Federation, April 7, 2016.

  • A. Vladimirov, Delay differential equation models of frequency swept laser light sources, International Conference on Structural Nonlinear Dynamics and Diagnosis (CSNDD'2016), University of Hassan II Casablanca, Marrakech, Morocco, May 24, 2016.

  • A. Vladimirov, Interaction of spatial and temporal cavity solitons in mode-locked lasers and passive cavities, 17th International Conference ``Laser Optics 2016'', June 27 - July 1, 2016, Saint Petersburg, Russian Federation, June 29, 2016.

  • A.G. Vladimirov, Distributed delay differential equation models in laser dynamics, Volga Neuroscience Meeting 2016, July 24 - 30, 2016, from Saint Petersburg to Nizhny Novgorod, Russian Federation, July 28, 2016.

  • A.G. Vladimirov, Interaction of temporal cavity solitons in driven fiber resonators and mode-locked lasers, International Tandem Workshop on Pattern Dynamics in Nonlinear Optical Cavities, August 15 - 19, 2016, Max-Planck-Institut für Physik komplexer Systeme, Dresden, August 15, 2016.

  • A.G. Vladimirov, Nonlinear dynamics of a frequency swept laser, Quantum Optics Seminar, Saint-Petersburg State University, Saint-Petersburg, Russian Federation, January 12, 2016.

  • M. Wolfrum, Emergence of collective behavior in coupled oscillator systems, Workshop ''Dynamics in Networks with Special Properties'', January 25 - 29, 2016, Mathematical Biosciences Institute (MBI), Columbus, USA, January 27, 2016.

  • M. Wolfrum, Emergence of collective behavior in coupled oscillator systems, Wednesdays@NICO, Northwestern University, Northwestern Institute on Complex Systems, Evanston, USA, January 20, 2016.

  • M. Wolfrum, Synchronization transitions in systems of coupled phase oscillators, Workshop on Synchronization and Oscillators with Generalized Coupling, April 20 - 22, 2016, University of Exeter, UK, April 21, 2016.

  • M. Wolfrum, Synchronization transitions in systems of coupled phase oscillators, Arbeitsgruppenseminar ``Chemische Physik fern vom Gleichgewicht'', Technische Universität München, Fachbereich Physik, March 18, 2016.

  • M. Wolfrum, Synchronization transitions in systems of coupled phase oscillators, Oberseminar Angewandte Mathematik, Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik und Informatik, June 22, 2016.

  • L. Lücken, Interplay of noise and synaptic plasticity in coupled neural oscillators, Workshop ``Dynamics and Stability of Interacting Nonlinear Oscillators and their Applications'', WIAS Berlin, Berlin, February 13, 2015.

  • S. Yanchuk, Delay-induced patterns in a two-dimensional lattice of coupled oscillators, 10th Colloquium on the Qualitative Theory of Differential Equations, July 1 - 4, 2015, University of Szeged, Bolyai Institute, Szeged, Hungary, July 4, 2015.

  • S. Yanchuk, How time delays influence dynamics, School of the International Research Training Group 1740 ``Dynamical Phenomena in Complex Networks'', July 20 - 21, 2015, Humboldt-Universität zu Berlin, Institut für Physik, Berlin, July 20, 2015.

  • U. Bandelow, Limitation for ultrashort solitons in nonlinear optical fibers by cusp formation, CLEO/Europe-EQEC 2015 Conference, June 21 - 25, 2015, München, June 25, 2015.

  • O. Omel'chenko, Chimera states in systems with control, EQUADIFF 2015, Minisymposium 3 ``Coupled Oscillator Systems and their Mean-Field Dynamics'', July 6 - 10, 2015, Lyon, France, July 9, 2015.

  • O. Omel'chenko, Creative control of chimera states, XXXV Dynamics Days Europe 2015, Minisymposium ``Controlling Complex Networks: Interplay of Structure, Noise, and Delay'', September 6 - 11, 2015, University of Exeter, Centre for Systems, Dynamics and Control, UK, September 9, 2015.

  • O. Omel'chenko, Paradoxes of the Kuramoto model, Seminar of the Department of Mathematics, Lomonosov Moscow State University, Russian Federation, November 25, 2015.

  • A.G. Vladimirov, Application of delay differential equations to the analysis of nonlinear dynamics in mode-locked lasers, Colloquium Nonlinear Sciences, Universität Münster, Center for Nonlinear Sciences, May 19, 2015.

  • M. Wolfrum, Chimera states with global feedback, Workshop on Control of Self-Organizing Nonlinear Systems, Wittenberg, September 14 - 16, 2015.

  • M. Wolfrum, Non-universal transitions to synchrony in the Sakaguchi--Kuramoto model, International Workshop on Dynamics of Coupled Oscillators: 40 Years of the Kuramoto Model, July 27 - 31, 2015, Max-Planck-Institut fúr Physik Komplexer Systeme, Dresden, July 30, 2015.

  • S. Amiranashvili, Elementary processes behind turbulent states in optical fibers, Weak Chaos and Weak Turbulence, February 3 - 7, 2014, Max-Planck-Institut für Physik komplexer Systeme, Dresden, February 5, 2014.

  • S. Amiranashvili, Extreme waves in optical fibers, Wave Interaction (WIN-2014), April 23 - 26, 2014, Johannes Kepler University, Linz, Austria, April 24, 2014.

  • S. Amiranashvili, Solitons who do not want to be too short, Workshop on Abnormal Wave Events (W-AWE2014), June 5 - 6, 2014, Nice, France, June 5, 2014.

  • U. Bandelow, Basic equations of classical soliton theory: Solutions and applications, BMS-WIAS Summer School ``Applied Analysis for Materials'', August 25 - September 5, 2014, Berlin Mathematical School, Technische Universität Berlin.

  • O. Omel'chenko, Bifurcation analysis of chimera states, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 13: Nonlocally Coupled Dynamical Systems: Analysis and Applications, July 7 - 11, 2014, Madrid, Spain, July 7, 2014.

  • O. Omel'chenko, Eckhaus scenario for partially coherent twisted states in arrays of coupled phase oscillators, XXXIV Dynamics Days Europe, September 7 - 12, 2014, Bayreuth, September 11, 2014.

  • O. Omel'chenko, Spike solutions to singularly perturbed elliptic problems, Workshop ``Modern Problems of Mathematical Physics'', November 28 - 29, 2014, Lomonosov Moscow State University, Russian Federation, November 28, 2014.

  • A.G. Vladimirov, Delay differential equations in laser dynamics, International Conference-School Hamiltonian Dynamics, Nonautonomous Systems, and Patterns in PDE's, December 10 - 15, 2014, Nishni Novgorod, Russian Federation, December 14, 2014.

  • M. Wolfrum, Stabilizing chimera states by feedback control, Colloquium ``Applications of Dynamical Networks'' of the Collaborative Research Center 910, Technische Universität Berlin, June 20, 2014.

  • S. Amiranashvili, Solitons that are too short in duration, International Workshop: Extreme Nonlinear Optics & Solitons, October 28 - 30, 13, WIAS Berlin, October 28, 2013.

  • O. Omel'chenko, Thermodynamic limit approach for bifurcation analysis of chimera states, Forschungsseminar ``Dynamische Systeme'', Freie Universität Berlin, January 10, 2013.

  • O. Omel'chenko, M. Wolfrum, Generalizing the Ott--Antonsen method for coupled phase oscillators, Dynamics Days Berlin-Brandenburg, October 1 - 2, 2013, Technische Universität Berlin, October 2, 2013.

  • O. Omel'chenko, Nonuniversal transitions to synchrony in the Sakaguchi--Kuramoto model, XXXIII Dynamics Days Europe, Minisymposium MS6 ``Collective Behavior in Networks of Oscillators'', June 3 - 7, 2013, Madrid, Spain, June 4, 2013.

  • O. Omel'chenko, Nonuniversal transitions to synchrony in the Sakaguchi-Kuramoto model, SIAM Conference on Applications of Dynamical Systems (DS13), May 19 - 23, 2013, Snowbird, USA, May 22, 2013.

  • O. Omel'chenko, Synchronization phenomena in large size systems of coupled oscillators, Bogolyubov Readings DIF-2013 ``Differential Equations, Theory of Functions and Their Applications'', June 23 - 30, 2013, Sevastopol, Ukraine, June 24, 2013.

  • M. Wolfrum, The Turing bifurcation on networks: Collective patterns and single differentiated nodes, International Conference on Dynamics of Differential Equations, March 16 - 20, 2013, Georgia Institute of Technology, Atlanta, USA, March 18, 2013.

  • M. Wolfrum, Chimera states: Patterns of coherence and incoherence in coupled oscillator systems, Dynamical Systems and Mathematical Biology Seminar, Georgia State University, Atlanta, USA, March 12, 2013.

  • M. Wolfrum, The Turing bifurcation on networks: Collective patterns and differentiated nodes, Kolloquium SFB 910, Technische Universität Berlin, Berlin, January 11, 2013.

  • M. Wolfrum, The Turing bifurcation on networks: Collective patterns and single differentiated nodes, Applied Dynamics Seminar, University of Maryland, Washington, USA, March 7, 2013.

  • S. Amiranashvili, Tiny waves we should never ignore, OSA -- The Optical Society, Topical Meeting ``Nonlinear Photonics'', June 17 - 21, 2012, Colorado Springs, USA, June 18, 2012.

  • R. Arkhipov, M. Radziunas, A. Vladimirov, Theoretical analysis of hybrid mode-locked quantum dot semiconductor lasers, International Conference ``Laser Optics 2012'', St. Petersburg, Russian Federation, June 25 - 29, 2012.

  • R. Arkhipov, M.V. Arkhipov, S.A. Pulkin, Numerical simulations of lasing without population inversion in two-level optically dense medium, International Conference ``Laser Optics 2012'', St. Petersburg, Russian Federation, June 25 - 29, 2012.

  • R. Arkhipov, Hybrid mode-locking in semiconductor quantum dot lasers: Simulation, analysis and comparison with experiments, ITN PROPHET Mid-Term Review Meeting, October 9 - 11, 2012, Paris, France, October 11, 2012.

  • R. Arkhipov, Numerical analysis of hybrid mode-locking in semiconductor quantum dot lasers, XIV All-Russian Scientific School-Seminar ``Wave Phenomena in Inhomogeneous Media'' (Waves-2012), Zvenigorod, Russian Federation, May 21 - 26, 2012.

  • R. Arkhipov, Spectral and temporal characteristics of resonant medium radiation excited at the superluminal velocity, International Symposium Advances in Nonlinear Photonics, September 23 - 27, 2012, Belarusian State University, Minsk, Belarus, September 26, 2012.

  • R. Arkhipov, The new principle of the all-optical streak camera based on ultrafast laser beam deflection by light-induced coherent photonic crystal, International Symposium Advances in Nonlinear Photonics, September 23 - 27, 2012, Belarusian State University, Minsk, Belarus, September 25, 2012.

  • R. Arkhipov, Theoretical investigation of hybrid mode-locking in two-section semiconductor quantum dot lasers, International Symposium Advances in Nonlinear Photonics, September 23 - 27, 2012, Belarusian State University, Minsk, Belarus, September 24, 2012.

  • I. Babushkin, Emission and control of coherent broad-band THz radiation using plasma-generating femtosecond light pulses, IPHT-Kolloquium, Institut für Photonische Technologien (IPHT), Jena, November 20, 2012.

  • O. Omel'chenko, Bifurcation analysis of chimera states, International Workshop: Coupled Networks, Patterns and Complexity, WIAS Berlin, November 21, 2012.

  • O. Omel'chenko, Chimera states: Spatiotemporal patterns of synchrony and disorder, Universität Hamburg, Department of Mathematics, November 12, 2012.

  • O. Omel'chenko, Coherence-incoherence patterns in systems of non-locally coupled phase oscillators, Statistical Physics and Nonlinear Dynamics & Stochastic Processes, Humboldt-Universität zu Berlin, Institut für Physik, Berlin, June 18, 2012.

  • O. Omel'chenko, Nonuniversal transitions to synchrony in the Sakaguchi--Kuramoto model, Seminar Applied Analysis, Humboldt-Universität zu Berlin, October 29, 2012.

  • O. Omel'chenko, What are chimera states, Westfälische Wilhelms-Universität Münster, Center for Nonlinear Science, November 6, 2012.

  • O. Omel'chenko, Coherence-incoherence patterns in systems of non-locally coupled phase oscillators, XXXII Dynamics Days Europe, September 2 - 7, 2012, University of Gothenburg, Sweden, September 4, 2012.

  • O. Omel'chenko, Synchronization transition in the Sakaguchi--Kuramoto model, 7th Crimean School and Workshop ``Emergent Dynamics of Oscillatory Networks'', May 20 - 27, 2012, Mellas, Crimea, Ukraine, May 22, 2012.

  • M. Wolfrum, Chimera states: Patterns of coherence and incoherence in coupled oscillator systems, Workshop ``Dynamics of Patterns'', December 16 - 21, 2012, Mathematisches Forschungsinstitut Oberwolfach, December 21, 2012.

  • M. Wolfrum, The Turing instability in irregular network systems, Jahrestagung der Deutschen Mathematiker-Vereinigung (DMV) 2012, Minisymposium ``Dynamical Systems'', September 18 - 20, 2012, Universität des Saarlandes, Fakultät für Mathematik und Informatik, Saarbrücken, September 20, 2012.

  • T. Girnyk, Two groups of globally coupled Kuramoto oscillators, Uni Potsdam, April 11, 2011.

  • O. Omelchenko, What does thermodynamic limit tell us about Chimera states?, SIAM Conference on Applications of Dynamical Systems (DS11), May 22 - 26, 2011, Society for Industrial and Applied Mathematics, Snowbird, Utah, USA, May 26, 2011.

  • V. Tronciu, Semiconductor lasers --- Key elements for chaos based communication systems, Università di Pavia, Ph.D. School of Electrical and Electronic Engineering and Computer Science, Italy, September 23, 2011.

  • M. Wolfrum, Mechanisms of semi-strong pulse interaction in the Schnakenberg model, Seminar z kvalitativnej teorie diferencialnych rovnic, Comenius University, Bratislava, Slovakia, November 10, 2011.

  • M. Wolfrum, Stability properties of equilibria and periodic solutions in systems with large delay, Equadiff 2011, August 1 - 5, 2011, University of Loughborough, UK, August 2, 2011.

  • M. Wolfrum, Stability properties of equilibria and periodic solutions in systems with large delay, The Sixth International Conference on Differential and Functional Differential Equations (DFDE 2011), August 17 - 21, 2011, Steklov Mathematical Institute, Moscow, Russian Federation, August 19, 2011.

  • J. Ehrt, Cascades of heteroclinic connections in viscous balance laws, 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, May 25 - 28, 2010, Technische Universität Dresden, May 27, 2010.

  • A.G. Vladimirov, Interaction of dissipative solitons and pulses in laser systems, Université Libre de Bruxelles, Optique Nonlinéaire Théorique, Belgium, April 21, 2010.

  • A.G. Vladimirov, Localized structures of light and their interaction, Imperial College London, Department of Applied Mathematics, UK, April 27, 2010.

  • A.G. Vladimirov, Nonlinear dynamics in lasers, Technische Universität Berlin, Institut für Festkörperphysik, March 24, 2010.

  • T. Girnyk, Multistability of twisted states in non-locally coupled Kuramoto-type models, Universität Potsdam, Institut für Physik und Astronomie, October 25, 2010.

  • T. Girnyk, Multistability of twisted states in non-locally coupled Kuramoto-type models, École Polytechnique Fédérale de Lausanne, Laboratory of Nonlinear Systems (EPFL-LANOS), Switzerland, November 17, 2010.

  • T. Girnyk, Stability of twisted states in repulsive Kuramoto models, Research Group Seminar, Freie Universität Berlin, research group ``Nonlinear Dynamics'', December 2, 2010.

  • M. Lichtner, Stability of delay differential equations with large delay, Dynamical System Seminar, Portsmouth University, Department of Mathematics, UK, March 17, 2010.

  • O.E. Omel'chenko, Coupling and motion of chimera states, Humboldt Kolleg Ukraine ``Mathematics and Life Sciences: Possibilities, Interlacements and Limits'', August 5 - 8, 2010, Kiev, Ukraine, August 7, 2010.

  • O.E. Omel'chenko, Dynamical properties of chimera states, Dynamics Days Europe, September 6 - 10, 2010, University of Bristol, Department of Engineering Mathematics, UK, September 6, 2010.

  • O.E. Omel'chenko, Moving chimera states, International Workshop ``Nonlinear Dynamics on Networks'', July 5 - 9, 2010, National Academy of Sciences of Ukraine, Kiev, July 6, 2010.

  • O.E. Omel'chenko, On the dynamical nature of chimera states, The 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, May 25 - 28, 2010, Technische Universität Dresden, May 25, 2010.

  • M. Wolfrum, Chimera states are chaotic transients, International Workshop ``Nonlinear Dynamics on Networks'', July 5 - 9, 2010, National Academy of Sciences of Ukraine, Kiev, July 6, 2010.

  • M. Wolfrum, Mechanisms of semi-strong pulse interaction in the Schnakenberg model, Emerging Topics in Dynamical Systems and Partial Differential Equations (DSPDEs'10), May 31 - June 4, 2010, International Center for Numerical Methods in Engineering, Barcelona, Spain, June 1, 2010.

  • M. Wolfrum, Mechanisms of semi-strong pulse interaction in the Schnakenberg model, Localized Structures in Dissipative Nonlinear Systems, October 18 - 20, 2010, WIAS, October 19, 2010.

  • M. Wolfrum, Routes to complex dynamics in a ring of unidirectionally coupled systems, Dynamics Days Europe 2010, September 6 - 10, 2010, University of Bristol, UK, September 7, 2010.

  • M. Wolfrum, Scaling properties of the spectrum for DDEs with large delay, Applied Maths Seminar, University of Exeter, Institute of Applied Mathematics, UK, November 22, 2010.

  • M. Wolfrum, Scaling properties of the spectrum for ODEs with large delay, 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, May 25 - 28, 2010, Technische Universität Dresden, May 25, 2010.

  • A.G. Vladimirov, Enhancement of interaction of dissipative solitons above self-pulsing instability threshold, CPNLW09 Soliton 2009 ``Solitons in Their Roaring Forties: Coherence and Persistence in Nonlinear Waves'', January 6 - 9, 2009, Nice University, Nice, France, January 8, 2009.

  • A.G. Vladimirov, Spontaneous motion of dissipative solitons under the effect of delay, Australasian Conference on Optics, Lasers and Spectroscopy and Australian Conference on Optical Fibre Technology in association with the International Workshop on Dissipative Solitons (ACOLS ACOFT DS 2009), November 29 - December 3, 2009, University of Adelaide, Australia, December 1, 2009.

  • A.G. Vladimirov, Strong enhancement of interaction of optical pulses induced by oscillatory instability, European Conference on Lasers and Electro-Optics and the XIth European Quantum Electronics Conference 2009 (CLEOtextsuperscript®/Europe -- EQEC 2009, Munich, June 14 - 19, 2009.

  • U. Bandelow, Semiconductor laser instabilities and dynamics (short course), 9th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD) 2009, September 14 - 18, 2009, Gwangju Institute of Science and Technology (GIST), Korea (Republic of), September 16, 2009.

  • M. Wolfrum, Asymptotic properties of the Floquet spectrum for delay differential equations with large delay, Seminario ISC, Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, Florence, Italy, April 30, 2009.

  • M. Wolfrum, Delay differential equations with large delay, Symposium ``Evolution Equations, Related Topics and Applications'', September 9 - 11, 2009, Helmholtz Zentrum München, September 9, 2009.

  • M. Wolfrum, The Eckhaus scenario in delay differential equation with large delay, International Workshop ``Trends in Bifurcation Analysis: Methods and Applications (TBA 2009)'', June 3 - 5, 2009, Milan, Italy, June 5, 2009.

  • J. Ehrt, Normally hyperbolic manifolds for viscous balance laws, Work Seminar ``Modelling, Analysis and Simulation'', Centrum voor Wiskunde & Informatica, Amsterdam, Netherlands, August 22, 2008.

  • J. Ehrt, Semi-strong interaction of pulses, Work Seminar ``Modelling, Analysis and Simulation'', Centrum voor Wiskunde & Informatica, Amsterdam, Netherlands, October 23, 2008.

  • S. Yanchuk, Bifurcations in lattices of unidirectionally coupled oscillators, Jour fixe, Graduiertenkolleg ``Analysis, Numerics, and Optimization of Multiphase Problems'', Humboldt-Universität zu Berlin, April 17, 2008.

  • S. Yanchuk, Destabilization in chains of coupled oscillators, Seminar of Work Group ``Neuromodulation'', Forschungszentrum Jülich, Institut für Neurowissenschaften und Biophysik, Teilinstitut Medizin, April 29, 2008.

  • U. Bandelow, Modeling and analysis of master-oscillator power-amplifier seminconductor lasers, University of Washington, Seattle, USA, October 16, 2008.

  • U. Bandelow, Short pulses in nonlinear optical fibers: Models and applications, Colloquium ``Nonlinear Dynamics in Complex Optical Systems'', Humboldt-Universität zu Berlin, Institut für Physik, June 19, 2008.

  • M. Wolfrum, Delay-differential equations with large delay, Seminar of the Working Group ``Dynamische Systeme'', Universität Hamburg, Department Mathematik, January 16, 2008.

  • M. Wolfrum, The Eckhaus scenario in delay-differential equations with large delay, Workshop ``Dynamics of Patterns'', December 14 - 20, 2008, Mathematisches Forschungsinstitut Oberwolfach, December 19, 2008.

  • S. Yanchuk, Eckhaus instability in systems with large delay, International Conference on Differential Equations (EQUADIFF 07), August 5 - 11, 2007, Vienna University of Technology, Austria, August 7, 2007.

  • S. Yanchuk, How size of a large system effects its dynamics?, European Conference on Complex Systems, October 1 - 6, 2007, Dresden, October 4, 2007.

  • U. Bandelow, Efficient modeling and analysis of dynamical effects in semiconductor laser devices, University of Nottingham, George Green Institute, UK, July 6, 2007.

  • U. Bandelow, Feedback enhanced modulation bandwidth, Dynamics Days Europe, University of Loughborough, UK, July 12, 2007.

  • U. Bandelow, Nichtlineare Effekte in Halbleiterlasern und optischen Fasern, Habilitandenkolloquium, Humboldt-Universität zu Berlin, Institut für Physik, April 17, 2007.

  • U. Bandelow, Semiconductor laser instabilities and dynamics (Short Course SC 0702), 7th International Conference ``Numerical Simulation of Optoelectronic Devices'' (NUSOD'07), University of Delaware, Newark, USA, September 25, 2007.

  • J. Ehrt, Slow-motion of multi-pulse solutions in reaction-diffusion systems by semistrong interaction, International Conference on Differential Equations (EQUADIFF 07), August 5 - 11, 2007, Vienna University of Technology, Austria, August 7, 2007.

  • M. Lichtner, Invariant manifold theorem for semilinear hyperbolic systems, EQUADIFF 07, August 5 - 11, 2007, Technische Universität Wien, Austria, August 7, 2007.

  • M. Wolfrum, Delay differential equations with large delay, Dynamical Systems Seminar, University of Minnesota, School of Mathematics, Minneapolis, USA, March 5, 2007.

  • S. Yanchuk, Amplitude equations for delay differential equations with large delay, Research Seminar Applied Analysis, Humboldt University of Berlin, Institute of Mathematics, April 27, 2006.

  • S. Yanchuk, Bifurcation theory for singularly perturbed systems with delay, National Academy of Sciences of Ukraine, Institute of Mathematics, Kiev, October 16, 2006.

  • S. Yanchuk, Bifurcations in systems with large delay, National Academy of Sciences of Ukraine, Institute of Mathematics, Kiev, May 29, 2006.

  • S. Yanchuk, Bifurcations in systems with long delay, Seminar of the Magnetoencephalography (MEG) Group, Research Center Jülich, Institute of Medicine, April 19, 2006.

  • S. Yanchuk, Hopf bifurcation for systems with large delay, Workshop ``Complex Dynamics and Delay Effects in Coupled Systems'', September 11 - 13, 2006, Humboldt-Universität zu Berlin, September 11, 2006.

  • S. Yanchuk, Typical instabilities in systems with large delay, 6th Crimean School and Workshops ``Nonlinear Dynamics, Chaos and Applications'', May 15 - 26, 2006, Yalta, Crimea, Ukraine, May 24, 2006.

  • U. Bandelow, Modeling and simulation of optoelectronic devices, Kick-off Workshop ``Materials in New Light'', Humboldt-Universität zu Berlin, Institut für Physik, Berlin, January 6, 2006.

  • U. Bandelow, Modellierung und Simulation optoelektronischer Bauelemente, Berliner Industriegespräche, Deutsche Physikalische Gesellschaft, Magnus-Haus, Berlin, September 6, 2006.

  • U. Bandelow, Simulation and analysis of spatio-temporal effects in complex laser structures, Kick-off Workshop ``Materials in New Light'', Humboldt-Universität zu Berlin, Institut für Physik, Berlin, January 6, 2006.

  • M. Lichtner, A spectral gap mapping theorem and smooth invariant center manifolds for semilinear hyperbolic systems, 6th AIMS International Conference on Dynamical Systems, Differential Equations & Applications, June 25 - 28, 2006, Université de Poitiers, France, June 28, 2006.

  • A. Vladimirov, Dynamics of light pulses in mode-locked lasers, 6th Crimean School and Workshops ``Nonlinear Dynamics, Chaos and Applications'', May 15 - 26, 2006, Yalta, Crimea, Ukraine, May 20, 2006.

  • A. Vladimirov, Laser dissipative solitons and their interaction, Minisymposium on Dissipative Solitons, WIAS, Berlin, April 20, 2006.

  • A. Vladimirov, Localized structures of light in laser systems and their weak interactions, Technische Universität Berlin, June 14, 2006.

  • A. Vladimirov, Nonlinear dynamics and bifurcations in multimode and spatially distributed laser systems, June 20 - 23, 2006, St. Petersburg State University, Russian Federation, June 20, 2006.

  • A. Vladimirov, Nonlinear dynamics in multimode and spatially extended laser systems, Moscow State University, Physics Faculty, Russian Federation, November 10, 2006.

  • A. Vladimirov, Transverse Bragg dissipative solitons in a Kerr cavity with refractive index modulation, Laser Optics Conference, June 26 - 30, 2006, St. Petersburg, Russian Federation, June 28, 2006.

  • M. Wolfrum, Describing a class of global attractors via symbol sequences, 6th AIMS International Conference on Dynamical Systems, Differential Equations & Applications, June 25 - 28, 2006, Université de Poitiers, France, June 28, 2006.

  • M. Wolfrum, Dynamics of chemical systems with mass action kinetics, Colloquium in Memory of Karin Gatermann, Universität Hamburg, Fachbereich Mathematik, January 7, 2006.

  • M. Wolfrum, Instabilities of laser systems with delay, 6th Crimean School and Workshops ``Nonlinear Dynamics, Chaos and Applications'', May 15 - 26, 2006, Yalta, Crimea, Ukraine, May 19, 2006.

  • M. Wolfrum, Systems of delay differential equations with large delay, Seminario do Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, Departamento de Matemática, Lisbon, Portugal, March 28, 2006.

  • A. Vladimirov, Interaction of dissipative solitons in laser systems, Ben Gurion University of the Negev, Department of Mathematics, Beer Sheva, Israel, November 17, 2005.

  • A. Vladimirov, Theoretical analysis of dynamical instabilities in a mode-locked semiconductor laser, Workshop ``Nonlinear Dynamics in Photonics'', May 2 - 4, 2005, WIAS, Berlin, May 3, 2005.

  • S. Yanchuk, Appearance of patterns in delay coupled laser arrays, Universität Potsdam, January 31, 2005.

  • S. Yanchuk, Bifurcations in systems with large delay, SFB 555 Symposium, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin, May 27, 2005.

  • S. Yanchuk, Instabilities of equililbria of delay-differential equations with large delay, ENOC 2005 (EUROMECH Nonlinear Oscillations Conference), August 7 - 12, 2005, Eindhoven, Netherlands, August 9, 2005.

  • S. Yanchuk, Normal forms for systems with large delay, National Academy of Sciences of Ukraine, Institute of Mathematics, Kiev, October 10, 2005.

  • S. Yanchuk, Properties of the Lang-Kobayashi model with large delay, Workshop ``Nonlinear Dynamics in Photonics'', May 2 - 4, 2005, WIAS, Berlin, May 2, 2005.

  • M. Nizette, A. Vladimirov, M. Wolfrum, D. Rachinskii, Delay differential equations for passive mode-locking, International Quantum Electronics Conference, München, June 12 - 17, 2005.

  • D. Turaev, S. Zelik, A. Vladimirov, Chaotic bound state of localized structures in the complex Ginzburg--Landau equation, Conference Digest ``Nonlinear Guided Waves and their Applications'', Dresden, September 6 - 9, 2005.

  • U. Bandelow, Analyse dynamischer Effekte in Optoelektronik und Photonik, Institutsseminar, Ferdinand-Braun-Institut für Höchstfrequenztechnik, Berlin, December 9, 2005.

  • V. Tronciu, Resonant coupling of a semiconductor laser to a Fabry-Perot resonator, Minisymposium ``Laser + Resonator'', WIAS, Berlin, February 17, 2005.

  • M. Wolfrum, Systems of delay differential equations with large delay, Otto-von-Guericke-Universität Magdeburg, Institut für Analysis und Numerik, June 14, 2005.

  • S. Yanchuk, Dynamics of two F2F coupled lasers: Instantaneous coupling limit, SPIE Photonics Europe 2004 Conference ``Semiconductor Lasers and Laser Dynamics'', April 27 - 30, 2004, Strasbourg, France, April 28, 2004.

  • S. Yanchuk, Intermittent synchronization in a system of coupled lasers, WIAS Workshop ``Synchronization and High-dimensional Chaos in Coupled Systems'', November 15 - 16, 2004, Berlin, November 15, 2004.

  • S. Yanchuk, Pattern formation in systems with large delay, Seminar ``Synchronization and Chaos'', National Academy of Sciences of Ukraine, Institute of Mathematics, Kiev, December 28, 2004.

  • S. Yanchuk, Singularly perturbed delay-differential equations. What do they have in common with ODEs and maps?, Seminar ``Nonlinear Oscillations'', National Academy of Sciences of Ukraine, Institute of Mathematics, Kiev, July 12, 2004.

  • U. Bandelow, 40 GHz mode-locked semiconductor lasers: Theory, simulation and experiments, Annual Meeting 2004 of the Optical Society of America (OSA) ``Frontiers in Optics'', October 10 - 14, 2004, Rochester, USA, October 11, 2004.

  • K.R. Schneider, Invariant manifolds for random dynamical systems with two time scales, Moscow State University, Faculty of Physics, Russian Federation, September 16, 2004.

  • K.R. Schneider, Invariante Mannigfaltigkeiten für zufällige dynamische Systeme mit schnellen und langsamen Variablen, Workshop GAMM-Fachausschuss "`Dynamik und Regelungstheorie"' und VDI/VDE-GMA-Ausschuss 1.40 "`Theoretische Verfahren der Regelungstechnik"', Universität Kassel, Regelungstechnik und Systemdynamik, March 8, 2004.

  • K.R. Schneider, Systeme mit schnellen und langsamen Variablen unter zufälligen Einwirkungen, Colloquium ``Singularly Disturbed Systems and Complex Dynamics'', June 16, 2004, Moscow State University, Faculty of Physics, Russian Federation, June 16, 2004.

  • U. Bandelow, Report on WIAS activities concerning COST Action 288, Kick-off Meeting for the Cost Action 288, COST TIST Secretariat, Brussels, Belgium, April 7, 2003.

  • U. Bandelow, Simulation of mode-locked lasers based on a distributed time-domain model, WIAS Workshop ``Dynamics of Semiconductor Lasers'', September 15 - 17, 2003, Berlin, September 17, 2003.

  • K.R. Schneider, Canard solutions of finite and infinite-dimensional dynamical systems, Moscow State University, Faculty of Physics, Russian Federation, October 1, 2003.

  • K.R. Schneider, Complete synchronization of nearly identical systems, WIAS Workshop ``Dynamical Systems, Synchronization, Lasers'', February 26 - 27, 2003, Berlin, February 26, 2003.

  • K.R. Schneider, Immediate and delayed exchange of stabilities, Belarussian State University, Institute for Mathematics, Minsk, November 18, 2003.

  • K.R. Schneider, Slow invariant manifold for a random dynamical system with two time-scales, EQUADIFF 2003, July 21 - 26, 2003, Hasselt, Belgium, July 25, 2003.

  • A. Vladimirov, Moving discrete solitons in multicore fibers and waveguide arrays, European Quantum Electronics Conference, June 22 - 27, 2003, München, June 25, 2003.

  • A. Vladimirov, Moving discrete solitons in multicore fibers and waveguide arrays, Conference dedicated to the 60th birthday of Prof. Paul Mandel, April 11 - 12, 2003, Université Libre de Bruxelles, Optique Nonlinéaire Théorique, Belgium, April 11, 2003.

  • M. Wolfrum, Attractors of semilinear parabolic equations on the circle, Dynamics of Structured Systems, December 14 - 20, 2003, Mathematisches Forschungszentrum Oberwolfach, December 16, 2003.

  • M. Wolfrum, Heteroclinic connections between rotating waves of scalar parabolic equations on the circle, EQUADIFF 2003, July 22 - 26, 2003, Hasselt, Belgium, July 23, 2003.

  • S. Yanchuk, Synchronization of two mutually coupled semiconductor lasers: Instantaneous coupling limit, WIAS Workshop ``Dynamics of Semiconductor Lasers'', September 15 - 17, 2003, Berlin, September 16, 2003.

  • S. Yanchuk, Synchronization phenomena in semiconductor laser, Sfb 555 Workshop ``Complex Nonlinear Processes'', September 11 - 13, 2003, Berlin, September 12, 2003.

  • S. Yanchuk, Complete synchronization of symmetrically coupled autonomous systems, EQUADIFF 2003, July 22 - 26, 2003, Hasselt, Belgium, July 25, 2003.

  • S. Yanchuk, Forced periodic frequency locking: Poincaré mapping approach, WIAS Workshop ``Dynamical Systems, Synchronization, Lasers'', February 26 - 27, 2003, Berlin, February 27, 2003.

  • S. Yanchuk, Synchronization of coupled autonomous systems, National Academy of Sciences of Ukraine, Institute of Mathematics, Kiev, April 21, 2003.

  • S. Yanchuk, Synchronization of two coupled Lang-Kobayashi systems, National Institute of Applied Optics, Florence, Italy, May 7, 2003.

  • S. Yanchuk, Synchronization problem in two-section semiconductor lasers, Forschungsseminar ``Angewandte Analysis'', Humboldt-Universität zu Berlin, Institut für Mathematik, July 7, 2003.

  • M. Wolfrum, Heteroclinic connections and order structures for scalar parabolic PDE, Instituto Superior Tecnico, Lisbon, Portugal, June 11, 2002.

  External Preprints

  • P. Kravetc, D. Rachinskii, A.G. Vladimirov, Pulsating dynamics of slow-fast population models with delay, Preprint no. arxiv.org:1601.06452, Cornell University Library, arXiv.org, 2016.
    Abstract
    We discuss a bifurcation scenario which creates periodic pulsating solutions in slow-fast delayed systems through a cascade of almost simultaneous Hopf bifurcations. This scenario has been previously associated with formation of pulses in a delayed model of mode-locked semiconductor lasers. In this work, through a case study of several examples, we establish that a cascade of Hopf bifurcations can produce periodic pulses, with a period close to the delay time, in population dynamics models and explore the conditions that ensure the realization of this scenario near a transcritical bifurcation threshold. We derive asymptotic approximations for the pulsating solution and consider scaling of the solution and its period with the small parameter that measures the ratio of the time scales. The role of competition for the realization of the bifurcation scenario is highlighted.

  • I. Omelchenko, O. Omel'chenko, P. Hövel, E. Schöll, Multi-chimera states in FitzHugh--Nagumo oscillators, Preprint no. arXiv:1212.3190, Cornell University Library, 2012.
    Abstract
    We demonstrate the existence of chimera states in a ring of identical oscillators described by FitzHugh-Nagumo equations with nonlocal coupling. This class of elements serves as a paradigmatic model in neuroscience, chemical oscillations, and nonlinear electronic circuits. Applying a phase-reduction technique we show that off-diagonal nonlocal coupling is a crucial factor for the appearance of chimera states, which consist of coexisting domains of coherent (phase-locked) and incoherent oscillators. Surprisingly, we find that for increasing coupling strength classical chimera states undergo transitions from one to multiple domains of incoherence. This additional spatial modulation is due to strong coupling interaction and thus cannot be observed in simple phase-oscillator models.

  • P. Perlikowski, S. Yanchuk, M. Wolfrum, A. Stefanski, P. Mosiolek, T. Kapitaniak, Routes to complex dynamics in a ring of unidirectionally coupled systems, Preprint no. 667, DFG Research Center sc Matheon, 2009.
    Abstract
    We study the dynamics of a ring of unidirectionally coupled autonomous Duffing oscillators. Starting from a situation where the individual oscillator without coupling has only trivial equilibrium dynamics, the coupling induces complicated transitions to periodic, quasiperiodic, chaotic, and hyperchaotic behavior. We study these transitions in detail for small and large numbers of oscillators. Particular attention is paid to the role of unstable periodic solutions for the appearance of chaotic rotating waves, spatiotemporal structures and the Eckhaus effect for a large number of oscillators. Our analytical and numerical results are confirmed by a simple experiment based on the electronic implementation of coupled Duffing oscillators.

  • M. Lichtner, M. Radziunas, Well posedness and smooth dependence for a semilinear hyperbolic system with nonsmooth data, Preprint no. 174, DFG Research Center sc Matheon, Technische Universität Berlin, 2004.