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

Kučera, Milan. "Bifurcation of periodic solutions to variational inequalities in $\mathbb {R}^\kappa $ based on Alexander-Yorke theorem." Czechoslovak Mathematical Journal 49.3 (1999): 449-474. <http://eudml.org/doc/30498>.

@article{Kučera1999,
abstract = {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.},
author = {Kučera, Milan},
journal = {Czechoslovak Mathematical Journal},
keywords = {bifurcation; periodic solutions; variational inequality; differential inequality; finite dimensional space; Alexander-Yorke theorem; bifurcation; periodic solutions; variational inequality; differential inequality; finite dimensional space; Alexander-Yorke theorem},
language = {eng},
number = {3},
pages = {449-474},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Bifurcation of periodic solutions to variational inequalities in $\mathbb \{R\}^\kappa $ based on Alexander-Yorke theorem},
url = {http://eudml.org/doc/30498},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Kučera, Milan
TI - Bifurcation of periodic solutions to variational inequalities in $\mathbb {R}^\kappa $ based on Alexander-Yorke theorem
JO - Czechoslovak Mathematical Journal
PY - 1999
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 49
IS - 3
SP - 449
EP - 474
AB - 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.
LA - eng
KW - bifurcation; periodic solutions; variational inequality; differential inequality; finite dimensional space; Alexander-Yorke theorem; bifurcation; periodic solutions; variational inequality; differential inequality; finite dimensional space; Alexander-Yorke theorem
UR - http://eudml.org/doc/30498
ER -