Deriving effective models for multiscale systems via evolutionary Γ-convergence

We discuss possible extensions of the recently established theory of evolutionary Γ-convergence for gradient systems to nonlinear dynamical systems obtained by perturbation of a gradient systems. Thus, it is possible to derive effective equations for pattern forming systems with multiple scales. Our applications include homogenization of reaction-diffusion systems, the justification of amplitude equations for Turing instabilities, and the limit from pure diffusion to reaction-diffusion. This is achieved by generalizing the Γlimit approaches based on the energy-dissipation principle or the evolutionary variational estimate.