Zero-Hopf bifurcations in 3-dimensional differential systems with no equilibria

Murilo R. Cândido; Jaume Llibre

doi:https://doi.org/10.1016/j.matcom.2018.03.008

Zero-Hopf bifurcations in 3-dimensional differential systems with no equilibria

Murilo R. Cândido, Jaume Llibre

Department of Mathematics

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

4 Citations (Scopus)

Abstract

© 2018 International Association for Mathematics and Computers in Simulation (IMACS) Recently sixteen 3-dimensional differential systems exhibiting chaotic motion and having no equilibria have been studied, and it has been graphically observed that these systems have a period-doubling cascade of periodic orbits providing a route to chaos. Here using new results on the averaging theory we prove that these systems exhibit, for some values of their parameters different to the ones having chaotic motion, either a zero-Hopf or a Hopf bifurcation, and graphically we observed that the periodic orbit starting in those bifurcations is at the beginning of the mentioned period-doubling cascade.

Original language	English
Pages (from-to)	54-76
Journal	Mathematics and Computers in Simulation
Volume	151
DOIs	https://doi.org/10.1016/j.matcom.2018.03.008
Publication status	Published - 1 Sep 2018

Keywords

Averaging theory
Periodic solutions
Quadratic polynomial differential system
Zero-Hopf bifurcation

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title = "Zero-Hopf bifurcations in 3-dimensional differential systems with no equilibria",

abstract = "{\textcopyright} 2018 International Association for Mathematics and Computers in Simulation (IMACS) Recently sixteen 3-dimensional differential systems exhibiting chaotic motion and having no equilibria have been studied, and it has been graphically observed that these systems have a period-doubling cascade of periodic orbits providing a route to chaos. Here using new results on the averaging theory we prove that these systems exhibit, for some values of their parameters different to the ones having chaotic motion, either a zero-Hopf or a Hopf bifurcation, and graphically we observed that the periodic orbit starting in those bifurcations is at the beginning of the mentioned period-doubling cascade.",

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AU - Cândido, Murilo R.

AU - Llibre, Jaume

PY - 2018/9/1

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N2 - © 2018 International Association for Mathematics and Computers in Simulation (IMACS) Recently sixteen 3-dimensional differential systems exhibiting chaotic motion and having no equilibria have been studied, and it has been graphically observed that these systems have a period-doubling cascade of periodic orbits providing a route to chaos. Here using new results on the averaging theory we prove that these systems exhibit, for some values of their parameters different to the ones having chaotic motion, either a zero-Hopf or a Hopf bifurcation, and graphically we observed that the periodic orbit starting in those bifurcations is at the beginning of the mentioned period-doubling cascade.

AB - © 2018 International Association for Mathematics and Computers in Simulation (IMACS) Recently sixteen 3-dimensional differential systems exhibiting chaotic motion and having no equilibria have been studied, and it has been graphically observed that these systems have a period-doubling cascade of periodic orbits providing a route to chaos. Here using new results on the averaging theory we prove that these systems exhibit, for some values of their parameters different to the ones having chaotic motion, either a zero-Hopf or a Hopf bifurcation, and graphically we observed that the periodic orbit starting in those bifurcations is at the beginning of the mentioned period-doubling cascade.

KW - Averaging theory

KW - Periodic solutions

KW - Quadratic polynomial differential system

KW - Zero-Hopf bifurcation

UR - https://ddd.uab.cat/record/199370

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DO - https://doi.org/10.1016/j.matcom.2018.03.008

M3 - Article

VL - 151

SP - 54

EP - 76

JO - Mathematics and Computers in Simulation

JF - Mathematics and Computers in Simulation

SN - 0378-4754

ER -