Periodic processes

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Periodic processes

Collaborator: D. Rachinskii

Cooperation with: M. Brokate (Technische Universität München), A.M. Krasnosel'skii, V.S. Kozyakin (Institute for Information Transmission Problems, Moscow, Russia), A.V. Pokrovskii, O. Rasskazov (University College Cork, Ireland)

Description:

Localization of periodic orbits is one of the basic problems in the analysis and control of dynamical systems. Using methods of topological degree theory, we have developed new analytic approaches to nonlocal problems on periodic solutions of systems with nonlinearities of saturation type, which are typical particularly for problems of laser dynamics.

We have studied bifurcations of periodic solutions from infinity and obtained estimates for the number of, and asymptotic formulas for, solution branches, [1, 2]. It was proved that generically there are two branches of forced periodic oscillations. However, many (any even number of) branches may appear in degenerate situations, which are standard for problems on subharmonic solutions. We have derived sufficient conditions for the existence of subharmonics (periodic solutions of multiple periods 2 $\pi$ n) with arbitrarily large amplitudes and periods for higher-order ordinary differential equations

L $\displaystyle \left(\vphantom{\frac{d}{dt},\lambda}\right.$ $\displaystyle {\frac{{d}}{{d t}}}$ , $\displaystyle \lambda$ $\displaystyle \left.\vphantom{\frac{d}{dt},\lambda}\right)$ x = f (x) + b(t),

(1)

$\minipage{0.37\textwidth}\Projektbild {\textwidth} {fig2_rach.eps}{ Frequency-lo... ... with the trajectory $\Gamma$ of a root of $L$}\label{fig2_rach_1} \endminipage$ depending on a scalar parameter $\lambda$ , with a 2 $\pi$ -periodic forcing term b and a nonlinearity f with saturation. This type of a subharmonic bifurcation from infinity occurs ([3]) whenever a pair of simple roots $\eta$ ( $\lambda$ ) $\pm$ $\xi$ ( $\lambda$ )i of the characteristic polynomial L crosses the imaginary axis at points $\pm$ $\alpha$ i with an irrational $\alpha$ . Under some further assumptions, we have estimated the parameter intervals of frequency locking, where subharmonics with amplitudes and periods increasing to infinity appear sporadically as the parameter approaches a bifurcation value $\lambda_{{0}}^{}$ . These assumptions relate the quality of the rational Diophantine approximations of $\alpha$ , the rate of convergence of the nonlinearity to its limits at infinity, and the smoothness of the forcing term. Actually, our results provide estimates for the width and length of the Arnold tongues in the problem of bifurcation of subharmonics from infinity (see Figure 1).

In [4], we applied our method to a Hopf bifurcation problem with a multiple resonance. Some numerical aspects of the problems were considered in [5].

A separate research direction was aimed at understanding the dynamics of systems where hysteresis plays an important role. Using the theory of closed systems with hysteresis operators as a framework for our studies, and concentrating mostly on periodic and cyclic processes, their stability and bifurcations, we have observed that hysteresis may be a source of several specific effects. For example, systems with hysteresis may have attractors consisting of a continuum of periodic or quasiperiodic orbits in generic situations. The results were tested with hysteresis models of different types.

We have studied the structure and stability of nontrivial connected clusters of periodic orbits that appear from an asymptotically stable periodic trajectory as a result of a hysteresis perturbation of the system, [6], as well as Hopf bifurcations of such attracting clusters from equilibria and from infinity, [7, 8]. A variety of dynamic and bifurcation scenarios (see, for example, Figure 2) was analyzed for the Duffing-type forced oscillator

x'' + x = sin( $\displaystyle \sqrt{{2}}$ t) + $\displaystyle \lambda$ P(x)(t)

(2)

with the hysteresis friction modeled by the Preisach operator P, on the basis of new topological approaches developed in [9], which combine analytic and rigorous computer-aided methods. Global stability of some complex hysteresis models, including that with the so-called ratchetting effect (unclosed hysteresis loops), was studied in [10, 11].

**Fig. 2:** Bifurcations of stable periodic orbits of the Poincaré map for Eq. (2) for different values of the parameter $\lambda$ : Two equilibria and two period-2 points (upper left); bifurcations of the equilibria to clusters of invariant curves (quasiperiodic orbits) and period doubling bifurcation of the period-2 orbits to period-4 orbits (upper right); further bifurcations of the periodic orbits to higher-order subharmonics (lower left); bifurcation to chaos (lower right)
$\makeatletter \@VierProjektbilderNocap[q]{0.25\textwidth}{fig2_rach_2.eps}{fig2_rach_3.eps}{fig2_rach_4.eps}{fig2_rach_5.eps} \makeatother$

References:

A.M. KRASNOSEL'SKII, D.I. RACHINSKII, Branching at infinity of solutions to equations with degeneration of multiplicity two, Doklady Math., 69(1) (2004), pp. 79-83.
, On a number of unbounded solution branches near the asymptotic bifurcation point, to appear in: Funct. Anal. Appl.
V.S. KOZYAKIN, A.M. KRASNOSEL'SKII, D.I. RACHINSKII, Subharmonic bifurcation from infinity, submitted.
A.M. KRASNOSEL'SKII, D.I. RACHINSKII, Remark on the Hopf bifurcation theorem, Math. Nachr., 272 (2004), pp. 95-103.
A.M. KRASNOSEL'SKII, A.V. POKROVSKII, D.I. RACHINSKII, On guaranteed estimates of convergence rate for one class of iteration procedures, Autom. Remote Control, 65(10) (2004), pp. 1635-1640.
M. BROKATE, A.V. POKROVSKII, D.I. RACHINSKII, Asymptotic stability of continual sets of periodic solutions to systems with hysteresis, WIAS Preprint no. 902, 2004.
M. BROKATE, D.I. RACHINSKII, Hopf bifurcations and simple structures of periodic solution sets in systems with the Preisach nonlinearity, WIAS Preprint no. 921, 2004.
D.I. RACHINSKII, On a bifurcation of stable large-amplitude cycles for equations with hysteresis, Autom. Remote Control, 65(12) (2004), pp. 1915-1937.
M. BROKATE, A.V. POKROVSKII, D.I. RACHINSKII, O. RASSKAZOV, Differential equations with hysteresis in examples, to appear in: The Science of Hysteresis, I. Mayergoyz, G. Bertotti, eds., Elsevier Science, 150 pages.
M. BROKATE, D.I. RACHINSKII, On global stability of the scalar Chaboche models, Nonlinear Anal. Real World Appl., 6 (2005), pp. 67-82.
, Global stability of Armstrong-Frederick models with periodic uniaxial inputs, to appear in: NoDEA Nonlinear Differential Equations Appl.