A dynamical systems perspective on spatial couplings in large chaotic systems

Presenter: Aditya Raj, Mechanical Engineering

Authors: A. Raj, M. R. Paul

Abstract: Complex spatiotemporal dynamics is exhibited by many natural systems. Examples include atmospheric and oceanic dynamics, nonequilibrium pattern formation, and the spread of disease in a highly mobile population. The dynamical systems approach enables us to study these intricate dynamics as a trajectory in state-space. The evolution of small perturbations is quantified by the covariant Lyapunov vectors (CLVs). Using CLVs, we explore the role of spatial couplings in spatiotemporal chaos. We find that  linear couplings, such as diffusion or the long-range connections of small-world networks, and the nonlinear coupling of convection, have strong influences on the shape of the Lyapunov spectra of coupled map lattices (CMLs). The coupling matrix is an essential element in linearly coupled systems. It can be used to predict the shape of the Lyapunov spectrum and the fractal dimension, and it strongly affects the spatial structure of CLVs. We also explore the role of weak mean flows on the chaotic dynamics of Rayleigh-B\'enard convection using the generalized Swift-Hohenberg equation and the CLVs.

Acknowledgements: Supported by NSF CMMI-2138055