Abstract
2003
Some important ideas froni classical control theory are introduced with the intention of applying them to chaotic dynamical systems, in particular the coupled logistic equations. The structure of this dissertation is such that a strong foundation in control theory is first established before introducing the coupled logistic map or the methods of control and targetting in chaotic systems.
In chapter 1 some aspects of classical control theory are reviewed.
Continuous- and discrete-time dynamical systems are introduced and the existence and
uniquendss criteria for the continuous case are explored via Lipschitz continuity.
The matrix form of an inhomogeneous linear differential equation is presented and several
properties of the associated transition matrix are discussed.
Several linear algebraic ideas, most notably the Cayley-Hamilton theorem, are employed to explore the important concepts of controllability and observability in linear systems.
The stabilisability problem is thoroughly investigated. Finally, the neighbourhood properties of continuous nonlinear dynamical systems with reference to controllability, stability and noise are established.
Chapter 2 places emphasis on canonical forms, pole assignments and state observers. The decomposition of a general system into distinct components is facilitated by the general structure theorem, which is proved. The pole placement problem is described and the correspondence between the stabilisability of a system and the placement of poles is noted by the use'of a socalled feedback matrix. Lastly, the notion of a state observer, with reference to some dynamic feedback law, is introduced.
The dynamics of the coupled logistic equations are studied in chapter 3. The fixed points of the map are calculated and the subsequent dynamical consequences explored. Using methods introduced in earlier chapters, the stability of the map is investigated. Using the so-called variational equations, the Lyapunov exponents are computed and used to classify, the motion of the system for the parameter values r and a. This chapter concludes with a discussion of the basins of attraction and critical curves associated with the coupled logistic equations.
It is in chapter 4 that the models for controlling chaos are instantiated. The famous Ott-Grebogi-
Yorke (OGY) method for controlling chaos is explained and related to the pole placement
problem, discussed previously. The theory is extended to study the control of periodic orbits with periods greater than one.