Research Group "Laser Dynamics"

The essential deviation of transport processes in complex systems from the classical behavior leads to the necessity of search for new approaches and scaling laws. One of the important directions is obtaining scaling laws that characterize transport in a stochastic magnetic field. In transport theory, one of the most widely applicable scaling is that for the root-mean-square deviation of randomly walking particle expressed in terms of the Hurst exponent H [1]. The purely empirical character of such an approach makes it necessary to search for quasi-diffusion equations that would provide a more rigorous justification of the scalings. One possible approach to solve this problem is to describe superdiffusion and subdiffusion by equations with fractional derivatives [2]. Originally, this approach was employed to analyze anomalous diffusion in conventional coordinate space. The next logical step was to describe nonlocal particle distribution functions in terms of both the fractional equations in velocity space [3] and the Boltzmann equation. A time-independent kinetic equation with the Fokker-Planck collision integral has been proposed that describes the effect of the stochastic magnetic field on the distribution function of energetic particles by means of the term with a fractional spatial derivative [4]. We consider a self-similar kinetic equation and establish the relationship between the Hurst exponent for anomalous spatial diffusion of high-energy solar energetic particles and the exponents characterizing the distribution function scaling [5-6]. Self-similar solutions to this equation have been analyzed to approximate power-law profiles of the plasma temperature and density. The ranges where the distribution function is significantly distorted have been determined, and the energies at which the distribution function changes its behavior have been estimated. With the parametric scalings obtained in this study, it is possible to use the results from measurements of the distribution function of energetic particles to analyze the mechanisms for turbulent transport in a stochastic magnetic field.

[1] Isichenko M B 1992 Rev. Mod. Phys. 64 961

[2] Bakunin O.G. 2003 Plasma Physics and Controlled Nucl. Fusion 45 1909

[3] Bakunin O.G. 2004 Reports on Progress in Physics 67

[4] Bakunin O G 2005 Physica A 345 1

[5] Bakunin O G 2005 J. Plasma Physics 71 756.

[6] Bakunin O G 2005 Chaos Solitons & Fractals 23 1703.

Weakly nonlinear theory of the Kuramoto transition in an ensemble of globally coupled oscillators in presence of additional time-delayed coupling terms is developed. It is shown that a linear delayed feedback not only shifts the transition point, but effectively changes the nonlinear terms near the transition. A purely nonlinear delayed coupling does not effect the transition point, but can reduce or enhance the amplitude of collective oscillations.

Arteriosclerosis is a common disease which severely influences human health. Arteriosclerosis characterized by stenosis which alter the overall characteristics of the vessels involved and wall shear stresses. In order to simulate the blood flow through stenosed coronary arteries nonlinear mathematical model is developed. The model takes into account arterial wall deformation under pulsatile flow condition. The flow of blood through the lumen region is governed by the continuity equation and the conservation of the area-averaged momentum. The velocity and pressure fields and shear stresses and deformation of the arterial wall are computed in a coupled manner through the use of the fluid-wall interaction condition. Simulation results indicate that the approach can generate an accurate estimation of the arterial blood pressure in real-time to noninvasive, continuous monitoring of arterial blood pressure for advanced cardiovascular diagnoses.

In this contribution, we study the conditions for the development of thermal and / or gravitational instabilities in the interstellar medium. We are especially interested in the question how radiative transfer effects can affect the stability of cold and dense interstellar molecular clouds. It turns out that the stability of the physical conditions - their density and kinetic temperature - are determined by the interplay of cosmic ray heating and cooling by the transitions of the abundant CO molecule and the IR emission of dust grains.

The results are of special interest in the context of star formation. For the formation of stars, the on average thin interstellar material has to be extremely compressed and the released gravitational energy has to escape from the system by radiative transfer processes. The critical cloud mass, i.e. the mass necessary to initiate a self gravitating collapse, depends therefore on the optical thickness of the medium being a measure for the probability that an emitted photon is reabsorbed within the collapsing cloud.

For the description of the radiative transfer, we use a stochastic model. We describe the variation of the turbulent velocity along each line of sight by a continuous Gauss Markov process. Caused by the stochastic nature of the underlying velocity field, the intensity of the radiation field itself becomes a stochastic variable. As a consequence, the ordinary radiative transfer equation has to be replaced by a generalised transfer equation of Fokker-Planck type. It turns out that the correlation length of the turbulent velocity field to the mean free path of photons has great influence on the intensity of the emitted lines and thereby on the cooling rates.

Using regulatorika (functioning of regulatory mechanisms) methodology for dynamical systems has allowed to develop the equations for study of a living system (ESLS) based on the functional-differential, functional and discrete equations [1,2].

From results of the qualitative analysis of model systems for ESLS equations follows, that functional state can have a varied nature: stable state, stable limit cycle, deterministic chaos (origin and development which was researched by calculations of Lyapunov number and Hausdorff rank) and break-down of solutions to the trivial attractor (one of the variants of ``black hole'' effect) [3-4].

The regularities of deterministic chaos final and scenario for beginning, development and terminating of ``black hole'' effect were analysed. Possibility for presence of only seven stable systems, which are in balance with external medium is determined, necessity of hierarchical organizations of development and existence of progressive evolutions of a living system are shown.

Study of genetic mechanisms of origin and developments of cancerous cells (using ESLS during grant of FPFI AS RUz 40-96 and 61-2000) has shown the possibility and successfulness of hypothesises about existence in genome the ADS (Autonomous Development Systems), which operate the early embryonic development. ADS activation in somatic cells can bring to cancer origin.

References

[1] Hidirov B.N. About one method of study regulatorica of living system, Voprosy kibernetiki, 128, Tashkent, 1984. P. 41-46 (in Russian).

[2] Hidirov B.N. Modelling of Regulation Mechanisms of Living System, Scientiae Mathematicae Japanicae. - Vol. 8 (2003). - P. 407-413.

[3] Hidirov B.N., Saidalieva M. Chaos research in autonomous regulating systems, Proceedings of the WCIS-2000. Tashkent, 2000. - P. 138-140.

[4] Hidirov B.N. Mathematical modeling of regulation mechanisms of living systems, International Scientific and Practical Conference 'Innovation-2001', Tashkent, 2001. P. 156-158.

On the basis of study methods for the living systems regulatorika [1] the differential-delay equations for analysis regulatory mechanisms in excitable media are constructed. Results of the qualitative study for equations and their model systems have shown presence the following regimes: rest, stable stationary state, Pouncare type limit cycles, dynamic chaos and ``black hole'' effect. Origin and development of irregular oscillations were valued by the calculation of Lyapunov's number near strange attractor [2-3]. Analysis of chaos region has shown that in chaos area there are ``r-windows'' - regions with regular solutions. Regularities of location, sizes and number of the ``r-windows'' are calculated. As example of biological excitable media we consider cardiac tissue and identification of chaos with an arrhythmia, ``black hole'' with sudden cardiac death has allowed studying regularities of cardiac activity at the norm and anomalies (disease, stress etc), condition determination for local correction in the chaos area for leaving in the norm area (in the area of Pouncare type limit cycles or ``r-windows''). Results of the investigations have used for analysis the mechanisms of portal hypertension origin (grant FPFI AS RUz 40-96) and for determination the mechanisms of origin and development of arrhythmia and sudden cardiac death (grant GKNT RUz 41/2000).

References.

[1] Hidirov B.N. About one method of study regulatorica of living system, Voprosy kibernetiki, 128, Tashkent, 1984, 41-46 (in Russian).

[2] Hidirova M.B. Modelling of Regulation Mechanisms of Cardiovascular Systems, Scientiae Mathematicae Japanicae. - Vol.8 (2003). - P. 423-428.

[3] Hidirova M.B. Biomechanics of cardiac activation: the simplest equations and modeling results, Russian Journal of Biomechanics, 2001.- Vol. 5, 2. - P. 95-103.

A nonlinear spatially extended system under the influence of noise and time-delayed feedback control is investigated. The model under consideration is a semiconductor superlattice which exhibits complex front dynamics associated with high-frequency current oscillations in the ideal noise-free case. We expand the model taking into account fluctuations, which we approximate by Gaussian white noise. Noise-induced front motion is reported, when the deterministic system is prepared in a stable fixed point slightly below a saddle-node bifurcation on a limit cycle. This is a global bifurcation, strongly related to excitability and the presence of two time scales in the system. Coherence resonance is confirmed by the non-monotonic relation between the regularity of oscillations and the noise intensity. The ability to control the properties of the system such as timescales and coherence, is very interesting in terms of applications. We therefore subject the system to a time-delayed feedback scheme, originally used to control deterministic chaotic dynamics, and demonstrate that the regularity and the time scales can be controlled by varying the control strength and time delay.

Torsionfree unstable periodic orbits cannot be stabilized by conventional time-delayed feedback control. The most simple example of such an orbit occurs at a subcritical Hopf bifurcation. Analytical and numerical investigations of an unstable van der Pol oscillator showed the successful control of such a torsionfree orbit by applying the idea of a nonlinear unstable feedback controller [1,2]. We succeeded to stabilize such a torsionfree unstable periodic orbit also in experiment by means of an electronic circuit realization of this model. The experiment showed that the basin of attraction of the controlled orbit is very small, so that practical application of this method might be difficult. In order to achieve robust control we modified the nonlinear control coupling. The advantages of our modification are discussed in comparison with the original idea.

[1] K. Pyragas, Phys. Rev. Lett. 86, 2265 (2001)

[2] K. Pyragas et al., Phys. Rev. E 70, 056222 (2004)

We present experimental results on varying degrees of synchronisation and the associated appearance of unstable dimension variability (UDV) in a physical system of coupled modified van der Pol electronic oscillators.

M.Muthuraman, J.Raethjen*, R.B.Govindan*, G.Deuschl*, U.Heute Institute for Circuit and System Theory and Dept. of Neurology*, University of Kiel

Phase shift between two signals from the Central Nervous System can emerge from local interneuronal connectivity patterns (eg. Central pattern generators). Shift in time rather implies a conduction delay between different neuronal centres. Thus distinguishing between these two processes has important implications. Delay estimation in broad band signals is easy by fitting a straight line to the coherent part of the phase spectrum. But this does not yield reliable results in narrow band signals. The maximising coherence method [1,2] uses the fact that due to a time delay between the two signals there is time misalignment which results in a reduction of coherence. In order to compensate for this reduction and to calculate the delay we artificially shift the two time series against each other in both directions. If there is a delay the coherence will increase and reach a maximum when the time shift equals the delay between the two signals. The phase shift between the two time series was taken from the phase spectrum which was estimated by the argument of the cross spectrum. The phase value is calculated for the frequency of coherence between the signals. These methods were applied to two narrow band pass filtered AR2 processes. In the model system delay and phase shift were introduced in between the two Ar2 processes. Changing the phaseshift does not have a relevant effect on the estimation of the specifically introduced delay. This was also tested for subjects with Parkinsonian tremor. In which the antagonistic muscles are activated in a reciprocal alternating pattern (phase shift of Pi). As expected we found that the activation pattern between antagonistic muscles were purely due to a phase shift. This intermuscular phase shift was also reflected in the cortico-muscular phase shift (between the contralateral EEG correlate of Parkinsonian tremor and forearm muscles EMG). There was a cortico-muscular phase reversal for the antagonistic muscles, whereas the cortico-muscular delay was between 8 and 30 ms in both directions for flexor as well as extensor muscles being in keeping with fast corticospinal transmission and feedback. Thus the method of maximising coherence for delay estimation does not detect the phase difference as a delay which ensures that we are able to effectively separate phase-shift and delay even in the narrow band coherent signals.

References:

1. Govindan RB, Raethjen J, Kooper F, Claussen JC, Deuschl G, Estimation of time delay by coherence analysis. Physica A 350:277-295

2. G.C. Carter, Proc.IEEE 75(1987) 236.

We study the influence of noise upon the nonlinear dynamics of molecular abundance patterns in a typical interstellar photon dominated region (PDR). PDRs are spatially extended regimes of the interstellar medium, the physics and especially chemistry of which are dominated by external UV radiation. In this contribution, the complex chemistry is described by a system of 38 coupled, nonlinear ODEs accounting for 437 photochemical and gas kinetic reactions, as well. The eigenvalue spectrum of this system is examined as a function of the strength of the incident UV radiation being the control parameter. Several sensitive parameter ranges, where noise induced oscillations can be excited, are identified. In dependence of the parameters of the external fluctuations (intensity, correlation time) we investigate the temporal behaviour of the noise-induced activity in this excitable medium and an explanation for the observed phenomena is given.

No abstract available.

A new kind of nonlinear nonequilibrium patterns - twisted spiral waves - is predicted for periodically forced oscillatory reaction-diffusion media. We show furthermore that, in such media, spatial regions with modified local properties may act as traps where propagating waves can be stored and released in a controlled way. Underlying both phenomena is the effect of the wavelength-dependent propagation reversal of traveling phase fronts, always possible when homogeneous oscillations are modulationally stable without forcing. The analysis is performed using as a model the complex Ginzburg-Landau equation, applicable for reaction-diffusion systems in the vicinity of a supercritical Hopf bifurcation.[1]

We consider two examples of realistic models describing reaction-diffusion systems with local oscillatory dynamics under the conditions of periodic forcing. Using the Krischer-Eiswirth-Ertl model for the catalytic CO oxidation on Pt(110) we demonstrate that phase front reversal can be expected under periodic variation of the CO partial pressure. Temperature heterogeneities on the Pt surface can be used to trap phase fronts. Another example for a more realistic model is the Oregonator which describes the light-sensitive BZ reaction. Our simulations of the Oregonator show that wavelength-dependent front propagation reversal is possible when the light intensity is varied periodically. It can be expected as well in the regime beyond the canard explosion, where the nature of the oscillations is relaxational rather than harmonic. Phase front traps can be realized by spatial variation of the forcing intensity.

[1] O. Rudzick and A. S. Mikhailov, Phys. Rev. Lett. 96, 018302 (2006)

We have proposed a quite general discrete model of active media by introducing a simple phase response curve interaction between pacemakers. This simplified model represents a network of pulse oscillators coupled by their response to internal depolarization of mutual stimulations. At first we have considered the macroscopic level and proposed the general model describing interaction of arbitrary large amount of oscillatory elements coupled globally. As specific cases we have examined a model of two bidirectionally interacting cardiac nodes and then in this model included an extra pacemaker, which can represent an external stimulater. Along with entrainment of the rhythm, complex behavior has been found. Next, we have moved to the microscopic level and represented cardiac nodes by one- and two-dimensional lattices of pulse oscillators coupled by the nearest neighbor principle. As a matter of fact on the basis of our unified model one can easily construct discrete distributed media of active elements, which interact via phase response curves.

It was investigated a cellular communities dynamics of à plant and animal organisms by the method of mathematical modeling with using a class of differential-delay equations [1,2]. Presence of the following regimes: rest, stationary state, Poincare type limit cycles, dynamic chaos and 'black hole' effect are determined.

Origin regularities of dynamic chaos, r-windowsregions and prediction problems for determination of destructive changes 'black hole' effect [3] are investigated.

Quantitative study of a regulation mechanisms of microorganisms, plants (at the consideration of a chlorella growing and a cotton wilt origin) and animal (at the analysis of an adaptive digestive system mechanisms) based on above mentioned method has shown importance of nonlinear phenomena in the functioning of a cellular communities [3,4].

Results of the work are used for the regularities analysis of cotton growing and development (grant FPFI AN RUz 41-98) and for the toolbox development for information technology in gene, cellular engineering and biotechnologies (grant DH-20.16 TsNT RUz).

References [1] Saidalieva M. Modeling of cellular communities //Voprosy kibernetiki, No 103. Tashkent, 1978. P. 89-103 (in Russian).

[2] Saidalieva M. Modelling of Regulation Mechanisms of Cellular Communities //Scientiae Mathematicae Japonicae, 2003. Vol. 58. T. 2 P. 463-469.

[3] Hidirov B.N., Saidalieva M. Chaos research in autonomous regulating systems //Proceedings of the WCIS - 2000. Tashkent, 2000. P. 138-140.

[4] Saidalieva M. Mathematical modeling of regulation mechanisms of cellular communities //International Scientific and Practical Conference Innovation - 2001. Tashkent, 2001. pp. 167-169.

Stars on the Asymptotic Giant Branch (AGB) are pulsating stars in a late stage of their stellar evolution. They suffer a massive mass loss via strong stellar winds. The winds are driven by radiation pressure on dust grains which condense out of the gas phase at distances of about 1-3 stellar radii from the star. Dynamical model calculations including hydrodynamics, thermodynamics, chemistry, dust formation and radiative transfer reveal a strong feedback between dust condensation and hydrodynamics via radiative transfer. The radiative backwarming of the outward moving dust shells heats the inner regions of the envelope, thus inhibiting further dust formation, until the shell has moved far enough out and the backwarming ceases. This process can lead to multiperiodicities in the hydrodynmical structure of the shell. The dust condensation itself shows a strong asymmetry with respect to temperature fluctuations, because the nucleation of new grains requires a much higher supersaturation than the growths of existing grains. We have therefore investigated the influence of temperature fluctuations on the dust condensation in these environments, which are dominated by strong non-linear coupling of hydrodynamics, radiation transport and dust condensation.

In a variety of excitable media, depending on the parameters different regimes of rotation have been observed for spiral waves including rigid rotation, meandering and hypermeandering. A proportional and a time-delayed feedback algorithm are elaborated to stabilize rigid rotation in a parameter range where it is unstable in the absence of feedback. As both control methods are non-invasive their application allows to determine the characteristic parameters of unstable rigid rotation. In our calculations the FHN model and the Oregonator model are taken as representative examples for excitable media. A latency time in the control loop shrinks the control domain for successful stabilization. We propose an effective method to overcome its destabilizing influence.

We analyze the phenomenon of complete synchronization in arrays of coupled, identical oscillators of Duffing type. We show that for a set of Duffing oscillators coupled by springs, an interesting phenomenon of alternately appearing 'windows' of synchronization and desynchronization can be observed. These 'windows' are visible on the bifurcation diagrams obtained versus coupling coefficient.

We study the constructive influence of noise upon the nonlinear dynamics of current density patterns in a semiconductor nanostructure, and its control by time delayed feedback methods. In particular, we investigate noise-induced pattern formation in a double barrier resonant tunnelling diode described by a nonlinear reaction-diffusion model.

The parameters of the system are fixed at values below a Hopf bifurcation where the only stable state of the deterministic system is a spatially inhomogeneous 'filamentary' steady state, and oscillating space-time patterns do not occur. We show that the addition of weak Gaussian white noise to the system gives rise to spatially inhomogeneous oscillations. As the noise intensity grows, the oscillations tend to become more and more spatially homogeneous, while simultaneously the temporal coherence of the oscillations decreases. We demonstrate that the application of a time delayed feedback loop allows one to control the temporal coherence and the time scales of the spatiotemporal patterns. The effects of the delayed feedback are explained by means of linear stability analysis of the spatially inhomogeneous steady state.

In the deterministic system a Hopf bifurcation leading to oscillating space-time patterns can also be induced by the delay.

There are some electronic systems with phase control, such as amplifier phase control systems (APC), originally designed for stabilization of phase and amplitude of radio signals in powerful high frequency amplifier tracts. APC's have studied very well in the literature as devices, intended for synchronizing and controlling of radio signal parameters, however practically have not been investigating as generators of chaotic oscillations. However, in our previous works we shown that chaos in APC has several especial properties: some these variants demonstrate unusual form of attractors (isoclin) in the phase space, intermittency near many fixed points, etc. APC's also have advantages in comparison with phase-locked loops (PLL) and other chaotic oscillators so that can be located just at the output cascades of powerful high frequency tracts and realize the chaotization of the input harmonic signal, carried to required power level and stability, for applying at radiating antenna of transmitter. The given work is devoted to the investigation of conditions for chaotic oscillation and methods of dynamical mode management in two amplifier phase control systems coupled via error signals of feedback loops. This type of chaotic generators is known to expand the zones of chaotic modes in parameter space for coupled PLL's and we have obtained analogous results for coupled APC's. We also consider that using of such interconnection can brings about both qualitatively new chaotic modes with accordance to single APC and elimination of chaotic oscillations at all, depending on the values of interconnection coefficients and the type of coupling: inphase or antiphase. It is defined that inphase coupling can leads both to new chaotic modes and to stabilization of regular oscillations. In the case of antiphase coupling the conservation of mode is observed, corresponding to original set-up of each subsystem, in the broad range of coupling coefficients values. These results allow us to give recommendations about setting the parameters of two coupled APC system in the chaotic modes of operation when using it as generator of radio frequency signals with chaotic phase and amplitude modulation. They are also can be useful providing the synchronization of several resonant amplifiers in high frequency tracts by means of phase control circuits.

We discuss the impact of resonant feedback from a Fabry Perot resonator (FPR) on both cw and self-pulsating operations of laser with saturable absorber. We start with a bifurcation analysis and the influence of parameters on stability of states. As shown previously, the locus of the external cavity modes in the case of feedback from FPR is a tilted eight with solitary laser mode (SLM) in its waist. With this in mind, with help of the DDE-BIFTOOL, we have found regions with instabilities along the tilted eight for self-pulsating laser and compared them with the cw laser. Next, we showed how the feedback from FPR affects the location of the Hopf bifurcation of SLM. Finally, we believe that our work provides a good basis for future studies and in particular provides some pointers for more detailed investigations of the influence of FPR feedback on the frequency of self-pulsations and realization of coherent control of pulses. These studies are mainly motivated by the application of FPR as a key element for all-optical implementation of Pyragas and Socolar delayed-feedback control method.

A system of averaged equations for two oscillators, similar to reported in [1], but with delay in coupling is considered. It is shown that for certain values of delay (``resonant'' delay), the system is associated with the Hamiltonian one and may be investigated by an approach, proposed in [2]. An analytical criterion of destruction the separatrix loops, originating from a saddle-focus stationary point is proposed basing on the Melnikov method.

[1] Y. Tsarin: Various orders of resonances and chaos in a two-mode quasilinear system, XXV Dynamics Days Europe, Germany, (2005).

[2] L. Lerman, Y. Umansky: About existence of separatrix loops in 4D systems closed to Hamiltonian integrable ones, Prikl.Math&Mech, 47, p.395 (1983)

Time-delayed feedback is a common technique used to stabilize unstable periodic orbits in nonlinear chaotic systems. Recently (Hövel and Schöll, 2005) it was applied to stabilize unstable stationary states. Here, their approach is simplified by dispensing with one term from the feedback and by applying the method to a single first order differential-difference equation. The complex eigenvalues of the characteristic equation are expressed exactly with the Lambert function. The boundary of the domain in the plane of control parameters (amplitude of forcing and time-delay) leading to asymptotic stability is also obtained analytically as is the line dividing between monotonous approach to the stable state and oscillatory convergence. The explicit expression of the former line is in agreement with the previously proven necessary and sufficient conditions for the boundary (Hayes, 1950). The method can be successfully applied to stabilize unstable stationary states in two or higher dimensional systems.

We study dynamics of FitzHugh-Nagumo neurons coupled with distributed time delays. Formation of stationary firing patterns in the network depends strongly on the network topology and initial conditions. In networks with randomly distributed delays, regimes with high periodicity on every element coexistent with low synchronization degree in the whole network are typical. Our simulations show, that decreasing the mean delay value and increasing its variance leads to lower synchronization degree in the network. We introduce the concept of 'leading center' to explain the formation of periodical regimes with low synchrony.

How do diverse dynamical patterns arise from the topology of complex networks? Here we investigate synchronisation dynamics on the cortico-cortical network of the cat by modelling each node (cortical area) with a sub-network of interacting excitable neurons. We find that in the biologically plausible regime the dynamics exhibits a hierarchical modular organisation, in particular, revealing functional clusters coinciding with the anatomical communities at different scales. Our results provide insights into the relationship between network topology and the functional organisation of complex brain networks. The next step is to implement delays into our model in order to investigate how it will affect the observed synchronisation between cortical areas and the presence of functional hierarchy.

It is believed that lethal cardiac pathology, ventricular fibrillation (VF), is produced by multiple wavelet re-entry (spiral waves in 2D and scroll waves in 3D). One of the ways to simplify the complex activity observed during experimental studies is to identify the tips of re-entrant waves on the heart surface, and this can be done by transforming the voltage distribution measured on the heart surface into phase. The tips of spiral waves are surrounded by tissue in all phases of the activation-recovery cycle, and hence are phase singularities (PSs). We have come to the conclusion that because the main factors responsible for turbulent activity are geometry, excitability etc., the PSs tracking is not sufficient for prediction of further dynamics of a medium. It is also necessary to compute invariant characteristics that are used in the theory of dynamical systems: Such an analysis has been carried out using a simplified ionic model of the cardiac action potential, so-called the three variable Fenton-Karma model, and it has revealed the following. Complex behaviour is not always the chaos. However, the knowledge whether an excitable medium possesses spatio-temporal chaotic dynamics is very important, since it allows to apply the phenomena of the phenomenon of the chaos suppression (i.e. defibrillation).

Thus, the obtained results point out the apparent need to reconsider some statements of contemporary research concerning dynamics of excitable media. Namely, one should take into account that the processes in excitable media may be more complex than a given set of re-entrant waves. The existing methods of the establishment of turbulent dynamics may be crucially incomplete to reveal the conditions at which it is possible to determine an antifibrillatory therapy. However, detection of such a therapy for several groups of patients would allow to decrease treatment costs (monitoring is always cheaper than treatment) and hence would significantly impact on economy. These results can have further applications by identifying the defibrillation protocol since it is possible to develop unpinning and other low-voltage defibrillation strategies.

We have tested low-voltage force suppressing (non-feedback point periodic stimulation of a prescribed shape) on the three variable Fenton-Karma model and FitzHugh-Nagumo like models and found that under some conditions re-entrant waves can be eliminated by mild stimulation (as a measure of suppression efficiency it was convenient to use PSs). Such low-power defibrillation is advantageous, since it does not require knowledge of both re-entry frequency (non-feedback method) and position. Moreover, in the case of complex turbulent pattern of activity, VF, all the rotating waves are suppressed simultaneously. Therefore, this new antifibrillation strategy should be realizable on practice.