Open Access Research

A bifurcation problem for a class of periodically perturbed autonomous parabolic equations

Mikhail Kamenskii1, Boris Mikhaylenko1 and Paolo Nistri2*

Author Affiliations

1 Department of Mathematics, Voronezh State University, Voronezh, 394006, Russia

2 Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, Siena, 53100, Italy

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Boundary Value Problems 2013, 2013:101  doi:10.1186/1687-2770-2013-101

Published: 23 April 2013

Abstract

The paper deals with the problem of the existence of a branch of T-periodic solutions originating from the isolated limit cycle of an autonomous parabolic equation in a Banach space when it is perturbed by a nonlinear T-periodic term of small amplitude.

We solve this problem by first introducing a novel integral operator, whose fixed points are T-periodic solutions of the considered equation and vice versa. Then we compute the Malkin bifurcation function associated to this integral operator and we provide conditions under which the well-known assumption of the existence of a simple zero of the Malkin bifurcation function guarantees the existence of the branch.

MSC: 35K58, 35B10, 35B20, 35B32.

Keywords:
autonomous parabolic equations; periodic perturbations; limit cycle; bifurcation; periodic solutions