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

Eder R. Aragão-Costa, Alexandre N. Carvalho, Pedro Marín-Rubio, Gabriela Planas

Abstract


We consider an autonomous dynamical system coming from a coupled system in cascade where the uncoupled part of the system satisfies that the
solutions 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.

Keywords


Dynamical system; nonlinear semigroup; attractor; gradient-like semigroup; Łojasiewicz-Simon inequality

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