On the Stability and Bifurcation Phenomena Associated with Dynamical Systems
These lectures are concerned with the bifurcation and stability behaviour of general dynamical systems. Emphasis is on multiple-parameter autonomous systems, and both static and dynamic bifurcations associated with such systems are explored analytically.
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K. Huseyin University of Waterloo, Waterloo, Ontario, Canada
ABSTRACT These lectures are concerned with the bifurcation and stability behaviour of general dynamical systems. Emphasis is on multiple-parameter autonomous systems, and both static and dynamic bifurcations associated with such systems are explored analytically. Thus, Hopf and degenerate Hopf bifurcations, as well as bifurcations into quasi-periodic motions on invariant tori are analysed. A brief discussion concerning the analysis of non-autonomous systems is presented in the last chapter.
INTRODUCTION Consider a dynamical system described by a finite set of n variables xi(i=1,2, .... ,n), where the xi = xi(t) represent the state of the system at time t, and the state evolves according to the ordinary differential equations ,t . X ,X , ... ,X n) -dxi-X(12 dt
I
A. N. Kounadis et al. (eds.), Nonlinear Stability of Structures © Springer-Verlag Wien 1995
(I)
K. Huseyin
170
Here, the ~ are assumed to be smooth functions of the variables x; and t - with continuous partial derivatives. This system is non-autonomous, and will be briefly considered in the last section. The emphasis, however, is on autonomous systems described by the vector differential equation dx = X(x) dt
(II)
where, the vector x represents the state variables. This system may exhibit various types of solutions. The salient features of the system are those that persist for a long time in the form of steady states, after the transients decay to zero as t ~ oo. The most prominent steady states are equilibria (stationary solutions) and periodic motions (eng. limit cycles). However, an autonomous system can also exhibit quasi-periodic motions, with more than one basic frequency, and chaotic motions. If the system is under the influence of one or more parameters, it can shift (or jump) from one solution to another at critical bifurcation points, as the parameters are varied slowly. Indeed, a particular steady-state may lose its stability when the parameters assume certain critical values, and the system bifurcates into other steady states. Here, the concept of stability is based on Lyapunov's definitions (see for example, Huseyin,1986), and bifurcations can take differenrforms depending on the nature of the vector field X(x). The aim of these lectures is to explore various bifurcation phenomena analytically and systematically. First, certain classes of systems are identified with regard to the nature of the vector field X(x) and the type of bifurcations that can be exhibited. The formulation and the method of analysis can then be chosen accordingly. Thus potential and non-potential systems, static and dynamic bifurcations, stability criteria, the effect of the structure of the Jacobian on the behaviour of the system are discussed. Hopf bifurcation and various degenerate Hopf bifurcations are analysed via an Intrinsic Harmonic Balancing technique. Bifurcations into invariant tori and the associated quasi-periodic motions are also discussed.
1. BASIC CONCEPTS AND CLASSIFICATIONS ABSTRACT
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