Transitions in Classical Dynamical Systems

We have been working to elucidate critical properties of transitions in classical dynamical systems by looking at 1-D and 2-D recurrence maps namely the circle map and Chirkov standard map.

For the 2-D standard map, we monitored the behavior of the system while transitioning from order to chaos by varying the coupling parameter. For the circle map, aka, climbing sine map, we focused on the properties of trajectories evolving in one cell or the ones transitioning to the neighboring cells for different order parameters. Different trajectories have been exhibiting periodic, chaotic, and diffusion-like behaviors as well as establishing cantor-set structures. We also attempted to engineer specific itineraries to construct confined and running orbits by establishing their cantor-set structure.

Added to that, we implemented numerical methods examining the parameters of both systems including diffusion coefficient, Lyapunov exponent, and rotation number.