Chaos and bifurcation theory

Bifurcation theory and the related theory of "chaos", pertains to dynamical systems, i.e. systems that evolve with time according to some rules, e.g. population evolution of species, climate models, traffic flow, etc. This theory aims to develop a qualitative theory of the behaviour of dynamical systems which change according to some input parameter values.

One class of dynamical systems are called "hybrid" systems, as such the evolution of such systems that is encoded in a function (usually real-valued) is a piecewise continuous function rather than a differentiable function. Many real-life systems can be modeled in such a way, from electric circuits to biological systems such as functioning of the heart.

Within such hybrid systems, a new class of bifurcation behaviour emerges which is qualitatively completely different from the classical dynamical systems with smooth evolution functions. These are called "border-collision" bifurcations. They appear when a fixed point or an equilibrium point crosses the boundary which separates the phase space into two pieces with different evolution equation parameters.

We develop the theory of such bifurcations in 3-dimensional hybrid systems. A typical attractor in such systems is shown in the figure below.

In joint work with S. Banerjee and M. Patra (IISER Kolkata, IN) we investigate bifurcations in 3-dimensional piecewise smooth maps which do not occur in lower dimensions.

Our recent paper describes the occurrence of Shilnikov-type dynamics for such systems and gives a necessary theoretical condition which guarantees this kind of dynamics.