Degenerate Fold-Hopf Bifurcations in a Rössler-Type System

G. Tigan; J. Llibre; L. Ciurdariu

doi:https://doi.org/10.1142/S0218127417500687

Degenerate Fold-Hopf Bifurcations in a Rössler-Type System

G. Tigan, J. Llibre, L. Ciurdariu

Department of Mathematics

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

2 Citations (Scopus)

Abstract

© 2017 World Scientific Publishing Company. We study the Hopf and the fold-Hopf bifurcations of the Rössler-type differential system [Equation presented here], with b≠0. We show that the classical Hopf bifurcation cannot be applied to this system for detecting the fold-Hopf bifurcation, which here is studied using the averaging theory. Our results show that a Hopf bifurcation takes place at the equilibrium (-ac/b,c/b,c/b) when c = a < 0. This Hopf bifurcation becomes a fold-Hopf bifurcation when c = a = 0.

Original language	English
Article number	1750068
Journal	International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Volume	27
Issue number	5
DOIs	https://doi.org/10.1142/S0218127417500687
Publication status	Published - 1 May 2017

Keywords

Rössler system
averaging theory
fold-Hopf bifurcation
periodic orbits
zero-Hopf bifurcation

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title = "Degenerate Fold-Hopf Bifurcations in a R{\"o}ssler-Type System",

abstract = "{\textcopyright} 2017 World Scientific Publishing Company. We study the Hopf and the fold-Hopf bifurcations of the R{\"o}ssler-type differential system [Equation presented here], with b≠0. We show that the classical Hopf bifurcation cannot be applied to this system for detecting the fold-Hopf bifurcation, which here is studied using the averaging theory. Our results show that a Hopf bifurcation takes place at the equilibrium (-ac/b,c/b,c/b) when c = a < 0. This Hopf bifurcation becomes a fold-Hopf bifurcation when c = a = 0.",

keywords = "R{\"o}ssler system, averaging theory, fold-Hopf bifurcation, periodic orbits, zero-Hopf bifurcation",

author = "G. Tigan and J. Llibre and L. Ciurdariu",

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AU - Tigan, G.

AU - Llibre, J.

AU - Ciurdariu, L.

PY - 2017/5/1

Y1 - 2017/5/1

N2 - © 2017 World Scientific Publishing Company. We study the Hopf and the fold-Hopf bifurcations of the Rössler-type differential system [Equation presented here], with b≠0. We show that the classical Hopf bifurcation cannot be applied to this system for detecting the fold-Hopf bifurcation, which here is studied using the averaging theory. Our results show that a Hopf bifurcation takes place at the equilibrium (-ac/b,c/b,c/b) when c = a < 0. This Hopf bifurcation becomes a fold-Hopf bifurcation when c = a = 0.

AB - © 2017 World Scientific Publishing Company. We study the Hopf and the fold-Hopf bifurcations of the Rössler-type differential system [Equation presented here], with b≠0. We show that the classical Hopf bifurcation cannot be applied to this system for detecting the fold-Hopf bifurcation, which here is studied using the averaging theory. Our results show that a Hopf bifurcation takes place at the equilibrium (-ac/b,c/b,c/b) when c = a < 0. This Hopf bifurcation becomes a fold-Hopf bifurcation when c = a = 0.

KW - Rössler system

KW - averaging theory

KW - fold-Hopf bifurcation

KW - periodic orbits

KW - zero-Hopf bifurcation

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