Bifurcation of periodic solutions to variational inequalities in $\mathbb{R}^\kappa$ based on Alexander-Yorke theorem

Kučera, Milan

Czechoslovak Mathematical Journal, Tome 49 (1999) no. 3, pp. 449-474.

Résumé

Variational inequalities \[ U(t) \in K, (\dot{U}(t)-B_\lambda U(t) - G(\lambda ,U(t)),\ Z - U(t))\ge 0\ \text{for all} \ Z\in \ K, \text{a.a.} \ t\in [0,T) \] are studied, where $K$ is a closed convex cone in $\mathbb{R}^\kappa $, $\kappa \ge 3$, $B_\lambda $ is a $\kappa \times \kappa $ matrix, $G$ is a small perturbation, $\lambda $ a real parameter. The assumptions guaranteeing a Hopf bifurcation at some $\lambda _0$ for the corresponding equation are considered and it is proved that then, in some situations, also a bifurcation of periodic solutions to our inequality occurs at some $\lambda _I \ne \lambda _0$. Bifurcating solutions are obtained by the limiting process along branches of solutions to penalty problems starting at $\lambda _0$ constructed on the basis of the Alexander-Yorke theorem as global bifurcation branches of a certain enlarged system.

Détail

Classification : 34A40, 34A60, 34C23, 34C25, 49J40, 58F14
Keywords: bifurcation; periodic solutions; variational inequality; differential inequality; finite dimensional space; Alexander-Yorke theorem