Linearization of Random Dynamical Systems

Abstract

At the end of the last century the French mathematician Henri Poincaré laid the foundation for what we call nowadays the qualitative theory of ordinary differential equations. Roughly speaking, this theory is devoted to studying how the qualitative behavior (e.g. the asymptotic behavior) of solutions to certain initial value problems changes as the initial condition is varied. In order to make this more explicit, consider the autonomous differential equation

$$ x = f(x), $$

(1.1)

where f : ℝ^d → ℝ^d is a C¹-mapping with f (x ₀) = 0 for some x ₀ ∈ ℝ^d. Obviously, the point x ₀ is a constant solution of (1.1). But what can be said about the behavior of solutions of (1.1) starting in some small neighborhood of this point? Of course one is tempted to first study the linearized equation

$$ x = Df(x_0 )x $$

(1.2)

near the origin, since it may be hoped that the nonlinear behavior of (1.1) near x ₀ will be “basically” the same.