Hopf bifurcation for some analytic differential systems in ℝ3 via averaging theory

Jaume Llibre; Clàudia Valls

doi:https://doi.org/10.3934/dcds.2011.30.779

Hopf bifurcation for some analytic differential systems in ℝ³ via averaging theory

Jaume Llibre, Clàudia Valls

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

11 Citations (Scopus)

Abstract

We study the Hopf bifurcation from the singular point with eigen-values a ε ± b i and c ε located at the origen of an analytic diffierential system of the form ẋ = f(x), where x ∈ ℝ3. Under convenient assumptions we prove that the Hopf bifurcation can produce 1, 2 or 3 limit cycles. We also characterize the stability of these limit cycles. The main tool for proving these results is the averaging theory of first and second order.

Original language	English
Pages (from-to)	779-790
Journal	Discrete and Continuous Dynamical Systems
Volume	30
Issue number	3
DOIs	https://doi.org/10.3934/dcds.2011.30.779
Publication status	Published - 1 Jul 2011

Keywords

Averaging theory
Diffierential systems in dimension
Hopf bifurcation

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abstract = "We study the Hopf bifurcation from the singular point with eigen-values a ε ± b i and c ε located at the origen of an analytic diffierential system of the form ẋ = f(x), where x ∈ ℝ3. Under convenient assumptions we prove that the Hopf bifurcation can produce 1, 2 or 3 limit cycles. We also characterize the stability of these limit cycles. The main tool for proving these results is the averaging theory of first and second order.",

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AB - We study the Hopf bifurcation from the singular point with eigen-values a ε ± b i and c ε located at the origen of an analytic diffierential system of the form ẋ = f(x), where x ∈ ℝ3. Under convenient assumptions we prove that the Hopf bifurcation can produce 1, 2 or 3 limit cycles. We also characterize the stability of these limit cycles. The main tool for proving these results is the averaging theory of first and second order.

KW - Averaging theory

KW - Diffierential systems in dimension

KW - Hopf bifurcation

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