Dynamical Stability and Bifurcation in B-Spaces

Abstract

Because of its great importance for science and numerical analysis, stability questions have been discussed already in a number of chapters of this volume and the three previous ones. In the present chapter we examine the following two important principles:

• (L) Linearization principle. The nonlinear differential equation has locally the same stability properties as the linearized differential equation.

• (B) Bifurcation principle. Loss of stability of an equilibrium point leads to bifurcation.

In case the differential equation can be integrated, the problem of stability presents no difficulty. It is important, however, to find methods which solve the stability problem independently of the integration.

Alexander Mihailovič Ljapunov (1892)

On passing through μ = 0 let us now assume that none of the characteristic exponents vanishes, but a conjugate pair crosses the imaginary axis. This situation commonly occurs in nonconservative mechanical systems, for example, in hydrodynamics. The following theorem asserts that, with this hypotheses, there is always a periodic solution in the neighborhood of the equilibrium point. In the literature, I have not come across this bifurcation problem. However, I scarcely think that there is anything essentially new in the above theorem. The methods have been developed by Poincaré perhaps 50 years ago, and belong today to the classical conceptual structure of the theory of periodic solutions.

Eberhard Hopf (1942)

Without the presence of stable phenomena, the world would pass into a state of complete chaos and its apparent structures would dissolve. There is no question that the discovery of interrelations in natural processes and its scientific description requires the existence of stable phenomena.

Herbert Beckert (1977)