Stability of dynamical systems

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Abstract
A new definition of the stability of ordinary differential equations is proposed as an alternative to structural stability. It is particularly aimed at dissipative nonlinear systems, including those with chaos or strange attractors. The definition is as follows. Given a vector field nu on an oriented manifold X, and given epsilon <0, let u be the steady state of the Fokker-Planck equation for nu with epsilon -diffusion. The existence, uniqueness and global attraction of u is proved in the case when X is compact (in the non-compact case a suitable boundary condition on nu is required for the existence of u). Vector fields are defined to be equivalent, or stable, according to whether their steady states are. A similar theory is developed for diffeomorphisms. The new definition has a number of advantages over structural stability. Stable systems are dense, and therefore most strange attractors are stable, including non-hyperbolic ones. The equivalence extends the Thom classification of gradient systems to non-gradient systems. The theory is closely related to applications, because the steady state u is an epsilon -smoothing of the measure on the attractors of the flow of nu , and therefore in numerical and physical experiments u can be used to model the data with epsilon -error.