Paper Info
Title
A Remark On Heteroclinic Bifurcations Near Steady State/Pitchfork Bifurcations
Abstract
We consider a bifurcation that occurs in some two-dimensional vector fields, namely a codimension-one bifurcation in which there is simultaneously the creation of a pair of equilibria via a steady state bifurcation and the destruction of a large amplitude periodic orbit. We show that this phenomenon may occur in an unfolding of the saddle-node/pitchfork normal form equations, although not near the saddle-node/pitchfork instability. By construction and analysis of a return map, we show that the codimension-one bifurcation emerges from a codimension-two point at which there is a heteroclinic bifurcation between two saddle equilibria, one hyperbolic and the other nonhyperbolic. We find the same phenomenon occurs in the normal form equations for the hysteresis/pitchfork bifurcation, in this case arbitrarily close to the instability, and show there are restrictions regarding the way in which such dynamics can occur near pitchfork/pitchfork bifurcations. These conclusions carry over to analogous phenomena in normal forms for steady state/Hopf bifurcations.
Year
DOI
Venue
2004
10.1142/S0218127404011752
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
Keywords
DocType
Volume
heteroclinic bifurcation, saddle-node/Hopf bifurcation, steady state/heteroclinic bifurcation, nonhyperbolic fixed points
Journal
14
Issue
ISSN
Citations 
11
0218-1274
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Vivien Kirk1225.61
Edgar Knobloch2276.68