Treffer: Averaging method for differential equations perturbed by dynamical systems

This is a large paper resulting from the author's Ph. D. thesis. Let \(Y_t: M\to M\) be a flow preserving a probability measure \(\mu\) on \(M\). Consider a differential equation \(dX/dt = f(X, Y_{t/\varepsilon})\) with an initial condition \(X(0)=x\), where \(f\) is uniformly bounded and uniformly Lipschitz continuous in the first argument, and \(\varepsilon >0\) a small parameter. Here \(X(t)\) is a slow variable and \(Y_{t/\varepsilon}\) is a fast variable. The main goal of the paper is to establish a convergence in distribution of the process \(X(t)\) to the solution \(W(t)\) of the averaging equation \(dW/dt = \int_M f(W, y) d\mu(y)\). The author proves several general theorems on such a convergence assuming that the flow \(Y_t\) (or the underlying base transformation, if \(Y_t\) is a suspension flow) has exponentially decaying multiple correlation functions. She also finds a limit distribution for the renormalized process \([X(t)- W(t)]/ \sqrt{\varepsilon}\) as \(\varepsilon \to 0\). General theorems are then applied to two examples -- ergodic toral automorphisms and Sinai billiards.