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Keywords:
bifurcation; periodic solutions; variational inequality; differential inequality; finite dimensional space; Alexander-Yorke theorem
Summary:
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.
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