Gradient-like nonlinear semigroups with infinitely many equilibria and applications to cascade systems

Abstract

We consider an autonomous dynamical system coming from a coupled system in cascade where the uncoupled part of the system satisfies that the solution comes from $-\infty $ and goes to $\infty $ to equilibrium points, and where the coupled part generates asymptotically a gradient-like nonlinear semigroup. Then, the complete model is proved to be also gradient-like. The interest of this extension comes, for instance, in models where a continuum of equilibrium points holds, and for example a Łojasiewicz-Simon condition is satisfied. Indeed, we illustrate the usefulness of the theory with several examples.