Qualitative techniques for bifurcation analysis of complex systems

Holmes, P. J. and J. E. Marsden

Ann. New York Acad. Sci., 316, (1979), 608-622

Abstract:

In this paper we consider systems whose dynamical behavior may be represented by an autonomous ordinary differential equation (ODE) with parameters,

$${\frac{{\mathrm{d}x}}{{\mathrm{d}t}}} \mu \equiv \mu$$ (1)

Here x is an element of a finite-dimensional vector space (say $\mathbb {R}$ⁿ ) or of a suitable Banach space of functions. In the latter case, (1) represents a partial differential equation (PDE). The control parameter $\mu$ $\in$ $\mathbb {R}$^m is supposed to vary slowly in comparison with the evolution rate of a typical solution x(t) of (1). Thus we treat (1) as an m -parameter family of ODEs. We are primarily interested in studying the qualitative changes that occur in the vector field or (semi-)flow defined by (1) as $\mu$ varies. The techniques used in the study of (1) draw on several fields, notably those of functional analysis and differentiable topology. In this brief paper we are only able to sketch general ideas and must therefore refer the reader to texts such as Chillingworth [5], and Marsden and McCracken [21] for background information and further details. Both texts contain a wealth of additional references.