Abstract
The coalescence of a Hopf bifurcation with a codimension-two cusp bifurcation of equilibrium points yields a codimension-three bifurcation with rich dynamic behavior. This paper presents a comprehensive study of this cusp-Hopf bifurcation on the three-dimensional center manifold. It is based on truncated normal form equations, which have a phase-shift symmetry yielding a further reduction to a planar system. Bifurcation varieties and phase portraits are presented. The phenomena include all four cases that occur in the codimension-two fold–Hopf bifurcation, in addition to bistability involving equilibria, limit cycles or invariant tori, and a fold–heteroclinic bifurcation that leads to bursting oscillations. Uniqueness of the torus family is established locally. Numerical simulations confirm the prediction from the bifurcation analysis of bursting oscillations that are similar in appearance to those that occur in the electrical behavior of neurons and other physical systems.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
