Chaos Suppression in Multi-Dimensional Systems

Professor Bogdan Epureanu
Dept. of Mechanical and Aerospace Engineering
Carleton University
Ottawa, Ontario, Canada  

Sponsored by the Dept. of Engineering Mechanics

Date:  Friday, May 5, 2000
Time:  3:30 p.m.
Place:  N213.3 Walter Scott Engineering Center


The underlying geometric structure of the standard Ott-Grebogi-Yorke (OGY) method for chaos suppression is analyzed. Some of the main mechanisms which may lead to the failure of the OGY method are discussed, and several alternate algorithms which overcome these problems are presented. The focus is primarily on the OGY technique applied to nonlinear flows, as distinct from nonlinear maps. Specifically, the case where the magnitude of the actuation varies in time within each period of the limit cycle to be stabilized is considered. The actuation varies in proportion to a given function referred to as a basis function. An algorithm for designing the optimal basis function is presented. A technique which generalizes the standard OGY scheme and may be applied to higher dimensional systems to stabilize both saddle nodes and fully unstable fixed points is presented also. The proposed method provides a powerful technique to investigate the controllability of nonlinear systems with OGY type controllers for chaos suppression. An optimal OGY method designed for larger dimensional systems and having improved convergence properties compared to the standard OGY controller is also presented. The analytical tools used in designing the multi-dimensional and optimal controllers may be used also to compute the linearized Poincare map and the sensitivity vector required for the implementation of the OGY method from experimental data. Unlike previous methods, the linearized map and the sensitivity vector are computed using only data collected over a single period of the limit cycle, therefore eliminating the long waiting period required by previously known methods before chaos suppression may be activated. Numerical examples for the well known Duffing and Van der Pol-Duffing oscillators are presented to illustrate the proposed techniques.