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Bifurcation of periodic solutions to variational inequalities in ℝ κ based on Alexander-Yorke theorem

Milan Kučera — 1999

Czechoslovak Mathematical Journal

Variational inequalities U ( t ) ∈ K , ( U ˙ ( t ) - B λ U ( t ) - G ( λ , U ( t ) ) , Z - U ( t ) ) ≥ 0 for all Z ∈ K , a.a. t ∈ [ 0 , T ) are studied, where K is a closed convex cone in ℝ κ , κ ≥ 3 , B λ is a κ × κ matrix, G is a small perturbation, λ a real parameter. The assumptions guaranteeing a Hopf bifurcation at some λ 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 λ I ≠ λ 0 . Bifurcating solutions are obtained by the limiting process along branches of solutions to penalty problems starting at λ 0 constructed...

Examples of bifurcation of periodic solutions to variational inequalities in ℝ κ

Milan Kučera — 2000

Czechoslovak Mathematical Journal

A bifurcation problem for variational inequalities U ( t ) ∈ K , ( U ˙ ( t ) - B λ U ( t ) - G ( λ , U ( t ) ) , Z - U ( t ) ) ≥ 0 for all Z ∈ K , a.a. t ≥ 0 is studied, where K is a closed convex cone in ℝ κ , κ ≥ 3 , B λ is a κ × κ matrix, G is a small perturbation, λ 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.

Bifurcations for a problem with jumping nonlinearities

Lucie Kárná;  Milan Kučera — 2002

Mathematica Bohemica

A bifurcation problem for the equation Δ u + λ u - α u + + β u - + g ( λ , u ) = 0 in a bounded domain in N with mixed boundary conditions, given nonnegative functions α , β ∈ L ∞ and a small perturbation g is considered. The existence of a global bifurcation between two given simple eigenvalues λ ( 1 ) , λ ( 2 ) of the Laplacian is proved under some assumptions about the supports of the functions α , β . These assumptions are given by the character of the eigenfunctions of the Laplacian corresponding to λ ( 1 ) , λ ( 2 ) .

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