Thomas Wanner

Linearization of Random Dynamical Systems

At the end of the last century the French mathematician Henri Poincare 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

Spinodal Decomposition¶for the Cahn–Hilliard–Cook Equation

x = f(x),

Coreduction Homology Algorithm for Regular CW-Complexes

(1.1) where f : ℝd → ℝd is a C1-mapping with f (x 0) = 0 for some x 0 ∈ ℝd. 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

Second phase spinodal decomposition for the Cahn-Hilliard-Cook equation

x = Df(x_0 )x

Perturbation of doubly periodic solution branches with applications to the Cahn-Hilliard equation

(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.

Maximum norms of random sums and transient pattern formation

This paper gives theoretical results on spinodal decomposition for the Cahn--Hillard equation. We prove a mechanism which explains why most solutions for the Cahn--Hilliard equation starting near a homogeneous equilibrium within the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius R. Namely, most solutions are driven into a region of phase space in which linear behavior dominates for much longer than expected.The Cahn--Hilliard equation depends on a small parameter