Symmetry and Localized Control of Extended Chaotic Systems

Citation

Grigoriev, Roman O. (1999) Symmetry and Localized Control of Extended Chaotic Systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/zxm1-n119. https://resolver.caltech.edu/CaltechTHESIS:08312022-180047419

Abstract

The present study is prompted by the failure of conventional chaos control theory to provide a practically sound algorithm for controlling the chaos in general spatially extended experimental systems. The primary reason for this failure is the presence of symmetry, which is a feature of most extended dynamical systems and which violates a number of assumptions of genericity made by conventional control theory. These assumptions can be relaxed, but at a price that increases with increasing symmetry of the target state. This price includes the larger number of independent control parameters that must be adjusted to steer the system towards the target trajectory, as well as the larger number of independent observables required to reconstruct the dynamics of an experimental system with symmetries.

We show that spatially extended chaotic systems can be controlled by monitoring and perturbing them at multiple spatial locations, or pinning sites, with separations determined by the noise in the system. We show that the arrangement of pinning sites must comply with constraints determined by the symmetry of the system in order to achieve control. We determine how the system can be forced from the spatiotemporally chaotic state into the controllable target state. Finally, we determine the maxim al distance between pinning sites and the maximal level of noise tolerated by a given arrangement of pinning sites for a model extended system.

Thesis Files