Abstract
We study the dynamical behaviour of a smooth vector field on a three-manifold near a heteroclinic network. Under some generic assumptions on the network, we prove that every path on the network is followed by a neighbouring trajectory of the vector field—there is switching on the network. We also show that near the network there is an infinite number of hyperbolic suspended horseshoes. This leads to the existence of a horseshoe of suspended horseshoes with the shape of the network.Our results are motivated by an example constructed by Field (1996 Lectures on Bifurcations, Dynamics, and Symmetry (Pitman Research Notes in Mathematics Series vol 356) (Harlow: Longman)), where we have observed, numerically, the existence of such a network.
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.
