Dynamical Systems

I have worked on the bifurcations and dynamics of high dimensional dynamical systems, both within the realm of smooth and piecewise smooth dynamical systems. Dimensionality of a dynamical system increases due to introduction of time delay, coupling two or more lower dimensional systems or interacting systems in general on a complex network.

The questions that I have been interested in are as follows:

1. In a high dimensional dynamical system how does the bifurcations unfold as parameters of the system change?

2. What is unique characteristics that emerge due to high dimensional characteristics of the system that is not readily observed in a lower dimensional system?

3. Are certain phenomenon observed in real physical problems of interests readily explainable in terms of a high dimensional dynamical system framework? 

I have investigated some of these questions in my research using toy models that include Henon mapping, Lozi mapping, Standard map (Kicked rotator), Rossler oscillator, Lorenz oscillator, and complex networks.

Examples of high - dimensional systems formed by coupling individual systems can exhibit phenomenon such as synchronization. A general formalism to study synchronization in such systems is the Master Stability Function (MSF) approach. I have worked on one aspect of the master stability analysis for coupled flows.

Coupled and forced/driven/perturbed dynamical systems:

As a detour to high dimensional systems, I have investigated quasiperiodically forced maps in the weakly dissipation regime. The problem is of practical interest since many practical systems are weakly dissipative and it would be interesting to know what happens to attractors of the system when forced. The investigation of this problem is made challenging/interesting due to multistability and chaotic transients.

Piecewise smooth dynamical systems

Piecewise smooth dynamical systems are smooth but not differentiable at a set of boundary points.  These systems occur in the study of electrical circuits due to switches and mechanical system that have impacting parts such as gears, or social systems where continuous change can induce discrete actions. Well known examples of piecewise smooth systems are tent map (in one dimension) and Lozi map (two dimensions). I studied the Lozi map by generalizing it to higher dimensions. Noteworthy were the emergence of hyperchaotic orbits, and convergence of fixed point dynamics in the parameter space as a function of dimensions (see . https://doi.org/10.1080/10236198.2022.2041625)