A SIMPLE APPROACH TO CALCULATION AND CONTROL OF UNSTABLE PERIODIC ORBITS IN CHAOTIC PIECEWISE-LINEAR SYSTEMS

This paper describes a simple method for calculating unstable periodic orbits (UPOs) and their control in piecewise-linear autonomous systems. The algorithm can be used to obtain any desired UPO embedded in a chaotic attractor; and the UPO can be stabilized by a simple state feedback control. A brief stability analysis of the controlled system is also given

Bifurcations, Chaos, Controlling and Synchronization of Certain Nonlinear Oscillators

In this set of lectures, we review briefly some of the recent developments in the study of the chaotic dynamics of nonlinear oscillators, particularly of damped and driven type. By taking a representative set of examples such as the Duffing, Bonhoeffer-van der Pol and MLC circuit oscillators, we briefly explain the various bifurcations and chaos phenomena associated with these systems. We use numerical and analytical as well as analogue simulation methods to study these systems. Then we point out how controlling of chaotic motions can be effected by algorithmic procedures requiring minimal perturbations. Finally we briefly discuss how synchronization of identically evolving chaotic systems can be achieved and how they can be used in secure communications.Comment: 31 pages (24 figures) LaTeX. To appear Springer Lecture Notes in Physics Please Lakshmanan for figures (e-mail: [email protected]

Two-parameter nonsmooth grazing bifurcations of limit cycles: classification and open problems

This paper proposes a strategy for the classification of codimension-two grazing bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a non-generic way. Several such codimension-one events have recently been identified, causing for example period-adding or sudden onset of chaos. Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincaré map from a neighbourhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the the grazing cycle is itself degenerate (e.g. non-hyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that have discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.

The role of singularities in chaotic spectroscopy

We review the status of the semiclassical trace formula with emphasis on the particular types of singularities that occur in the Gutzwiller-Voros zeta function for bound chaotic systems. To understand the problem better we extend the discussion to include various classical zeta functions and we contrast properties of axiom-A scattering systems with those of typical bound systems. Singularities in classical zeta functions contain topological and dynamical information, concerning e.g. anomalous diffusion, phase transitions among generalized Lyapunov exponents, power law decay of correlations. Singularities in semiclassical zeta functions are artifacts and enters because one neglects some quantum effects when deriving them, typically by making saddle point approximation when the saddle points are not enough separated. The discussion is exemplified by the Sinai billiard where intermittent orbits associated with neutral orbits induce a branch point in the zeta functions. This singularity is responsible for a diverging diffusion constant in Lorentz gases with unbounded horizon. In the semiclassical case there is interference between neutral orbits and intermittent orbits. The Gutzwiller-Voros zeta function exhibit a branch point because it does not take this effect into account. Another consequence is that individual states, high up in the spectrum, cannot be resolved by Berry-Keating technique.Comment: 22 pages LaTeX, figures available from autho

Wild oscillations in a nonlinear neuron model with resets: (II) Mixed-mode oscillations

This work continues the analysis of complex dynamics in a class of bidimensional nonlinear hybrid dynamical systems with resets modeling neuronal voltage dynamics with adaptation and spike emission. We show that these models can generically display a form of mixed-mode oscillations (MMOs), which are trajectories featuring an alternation of small oscillations with spikes or bursts (multiple consecutive spikes). The mechanism by which these are generated relies fundamentally on the hybrid structure of the flow: invariant manifolds of the continuous dynamics govern small oscillations, while discrete resets govern the emission of spikes or bursts, contrasting with classical MMO mechanisms in ordinary differential equations involving more than three dimensions and generally relying on a timescale separation. The decomposition of mechanisms reveals the geometrical origin of MMOs, allowing a relatively simple classification of points on the reset manifold associated to specific numbers of small oscillations. We show that the MMO pattern can be described through the study of orbits of a discrete adaptation map, which is singular as it features discrete discontinuities with unbounded left- and right-derivatives. We study orbits of the map via rotation theory for discontinuous circle maps and elucidate in detail complex behaviors arising in the case where MMOs display at most one small oscillation between each consecutive pair of spikes