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.
A bifurcation problem for variational inequalities \[ U(t) \in K, (\dot{U}(t)-B_\lambda U(t) - G(\lambda ,U(t)),\ Z - U(t))\ge 0\ \text{for} \text{all} \ Z\in K, \text{a.a.} \ t \ge 0 \] is 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 main goal of the paper is to simplify the assumptions of the abstract results concerning the existence of a bifurcation of periodic solutions developed in the previous paper and to give examples in more than three dimensional case.