# 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

, Volume 17 (2) – May 2, 2017
16 pages

/lp/springer_journal/analysis-of-degenerate-fold-hopf-bifurcation-in-a-three-dimensional-b6xN2JhRVx
Publisher
Springer Journals
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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations