On the limit cycles of the Floquet differential equation

Jaume Llibre; Ana Rodrigues

doi:10.3934/dcdsb.2014.19.1129

Article Contents

2014, Volume 19, Issue 4: 1129-1136. Doi: 10.3934/dcdsb.2014.19.1129

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On the limit cycles of the Floquet differential equation

Jaume Llibre^1, and
Ana Rodrigues^2,

1.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia
2.
College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QF

Received: March 31, 2012

Revised: January 31, 2014

Published: May 2014

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Abstract

We provide sufficient conditions for the existence of limit cycles for the Floquet differential equations $\dot {\bf x}(t) = A{\bf x}(t)+ε(B(t){\bf x}(t)+b(t))$, where ${\bf x}(t)$ and $b(t)$ are column vectors of length $n$, $A$ and $B(t)$ are $n\times n$ matrices, the components of $b(t)$ and $B(t)$ are $T$--periodic functions, the differential equation $\dot {\bf x}(t)= A{\bf x}(t)$ has a plane filled with $T$--periodic orbits, and $ε$ is a small parameter. The proof of this result is based on averaging theory but only uses linear algebra.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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