Analysis of Degenerate Fold–Hopf Bifurcation in a Three-Dimensional Differential System

Analysis of Degenerate Fold–Hopf Bifurcation in a Three-Dimensional Differential System In this work we study degenerate fold–Hopf bifurcation of codimension three in a differential system of the form $$\begin{aligned} {\dot{x}}=-x-y-z,\ {\dot{y}}=x+ay+bxy,\ {\dot{z}}=bx-cz. \end{aligned}$$ x ˙ = - x - y - z , y ˙ = x + a y + b x y , z ˙ = b x - c z . We prove that the classical theory of the fold–Hopf bifurcation cannot be applied for studying the system’s behavior. However, using tools from averaging theory we prove the existence of periodic orbits generated by this bifurcation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Qualitative Theory of Dynamical Systems Springer Journals

Analysis of Degenerate Fold–Hopf Bifurcation in a Three-Dimensional Differential System

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer International Publishing
Subject
Mathematics; Mathematics, general; Dynamical Systems and Ergodic Theory; Difference and Functional Equations
ISSN
1575-5460
eISSN
1662-3592
D.O.I.
10.1007/s12346-017-0241-4
Publisher site
See Article on Publisher Site

Abstract

In this work we study degenerate fold–Hopf bifurcation of codimension three in a differential system of the form $$\begin{aligned} {\dot{x}}=-x-y-z,\ {\dot{y}}=x+ay+bxy,\ {\dot{z}}=bx-cz. \end{aligned}$$ x ˙ = - x - y - z , y ˙ = x + a y + b x y , z ˙ = b x - c z . We prove that the classical theory of the fold–Hopf bifurcation cannot be applied for studying the system’s behavior. However, using tools from averaging theory we prove the existence of periodic orbits generated by this bifurcation.

Journal

Qualitative Theory of Dynamical SystemsSpringer Journals

Published: May 2, 2017

References

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