Hopf bifurcation in higher dimensional differential systems via the averaging method

Jaume Llibre and Xiang Zhang

Vol. 240 (2009), No. 2, 321–341

DOI: 10.2140/pjm.2009.240.321

Abstract

We study the Hopf bifurcation of 𝒞³ differential systems in ℝⁿ showing that l limit cycles can bifurcate from one singularity with eigenvalues ±bi and n− 2 zeros with l ∈{0,1,…,2ⁿ⁻³}. As far as we know this is the first time that it is proved that the number of limit cycles that can bifurcate in a Hopf bifurcation increases exponentially with the dimension of the space. To prove this result, we use first-order averaging theory. Further, in dimension 4 we characterize the shape and the kind of stability of the bifurcated limit cycles. We apply our results to certain fourth-order differential equations and then to a simplified Marchuk model that describes immune response.

Keywords

limit cycles, generalized Hopf bifurcation, averaging theory

Mathematical Subject Classification 2000

Primary: 34C23, 34C29, 37G15

Milestones

Received: 18 November 2007

Accepted: 6 February 2009

Published: 4 March 2009