Topological degree in analysis of chaotic behavior in singularly perturbed systems

Abstract

A scheme of applying topological degree theory to the analysis of chaotic behavior in singularly perturbed systems is suggested. The scheme combines one introduced by Zgliczynski [Topol. Methods Nonlinear Anal. 8, 169 (1996)] with the method of topological shadowing, but does not rely on computer based proofs. It is illustrated by a three-dimensional system with piecewise linear slow surface. This approach, when applicable, guarantees abundance of periodic orbits with arbitrarily large periods, each of which is a canard-type trajectory: at first it passes along, and close to, an attractive part of the slow surface of the singularly perturbed system and then continues for a while along the repulsive part of the slow surface. These periodic trajectories are robust in a topological sense with respect to small disturbances in the right-hand sides of the system under consideration, but typically not stable in the Lyapunov sense. Methods of localization of such periodic trajectories are briefly discussed, and numerical examples of localizations are given. The periodic trajectories that are useful from the applications point of view can be stabilized via an appropriate feedback control, for instance, the Pyragas control.

Chaos in a seasonally perturbed SIR model: avian influenza in a seabird colony as a paradigm

“…Computer-assisted applications of the method were used to prove the existence of chaotic trajectories for the extended Korteweg-de Vries-Burgers wave equations (Cox et al 2005), for the Kaldor model of the trade cycle with a hysteretic nonlinearity (McNamara and Pokrovskii 2006) and for a piecewise linear oscillator (Pokrovskii et al 2007). A proof of the existence of chaos in a singularly perturbed system using the method, which was not computer-aided, was provided by Pokrovskii and Zhezherun (2008). While it is well known that seasonally perturbed population-pathogen systems can exhibit a variety of dynamics, including chaotic behaviour (Aron and Schwartz 1984;Dushoff et al 2004;Earn et al 2000;Keeling et al 2001;Olsen and Schaffer 1990), it is a highly non-trivial task to provide a proof of the existence of chaos in such a system.…”

“…The topological hyperbolicity method can be applied to prove chaos in the Smale sense, i.e., to show the existence of a correspondence between the left shift mapping on the set of symbolic sequences and a restriction of some fixed iterate of the translation operator of the dynamical system under consideration to an invariant set. The method has been applied to prove the existence of chaotic trajectories for a wide variety of strongly nonlinear systems (Cox et al 2005;McNamara and Pokrovskii 2006;Pokrovskii et al 2007;Pokrovskii and Zhezherun 2008). We will illustrate the theory using the seasonally perturbed SIR model of the seabird-avian influenza system.…”

A corollary of the Poincaré–Bendixson theorem and periodic canards

“…The construction of the region D in the case of a headless canard is the same as in the first example, and in the case of a headed canard is explained in The results of this paper are relevant to the use of topological degree in analysis of canards in multi-dimensional systems [5]; they were formulated in our preprint [6]. …”

“…We say that at a = 0 system (4) has a family of periodic canards of magnitude α > 0, if to any small ε > 0 one can correspond a ε and a periodic solution (x ε,a ε (t), y ε,a ε (t)) of the systemẋ = y, εẏ = −x + F (y + a ε ), such that max x ε,a ε (t): y ε,a ε (t) = 0 = α. (5) In our case a periodic solution may visit the upper half-plane y > 0 only traveling along the repulsive branch of the slow curve. Thus, this definition is consistent with the informal explanation given above.…”

Alexei Vadimovich Pokrovskii, 1948–2010

“…One means Alexei used to establish the UCC Department of Applied Mathematics as a centre of research excellence in applied mathematics and dynamics of hysteretic systems on the international scene was the Multi-rate Processes and Hysteresis conference series. The conference originated from his idea to explore the links between the methods of the theory of multi-rate systems and the theory of systems with hysteresis [52,53]; Van der Pol relaxation oscillations is one classical fundamental example of such links. After initial discussions with Michael P. Mortell and with Robert E. O'Malley and Vladimir Andreevich Sobolev, two world renowned experts in the analytic and geometric theory of singularly perturbed systems, with whom Alexei met at the Industrial Mathematics Congress in Edinburgh in 1999, a pilot workshop was organised in Cork in 2001.…”