Stability in simple heteroclinic networks in ${\mathbb{R}}^4$

Abstract

We describe all heteroclinic networks in ${\mathbb{R}}^4$ made of simple heteroclinic cycles of types $B$ or $C$, with at least one common connecting trajectory. For networks made of cycles of type $B$, we study the stability of the cycles that make up the network as well as the stability of the network. We show that even when none of the cycles has strong stability properties the network as a whole may be quite stable. We prove, and provide illustrative examples of, the fact that the stability of the network does not depend {\em a priori} uniquely on the stability of the individual cycles.

Cited by in corpus (10)

• Stability of cycling behaviour near a heteroclinic network model of Rock-Paper-Scissors-Lizard-Spock
• Heteroclinic Dynamics of Localized Frequency Synchrony: Stability of Heteroclinic Cycles and Networks
• A trio of heteroclinic bifurcations arising from a model of spatially-extended Rock-Paper-Scissors
• Almost complete and equable heteroclinic networks
• Switching in heteroclinic networks
• Construction of heteroclinic networks in ${\mathbb{R}}^4$
• Simple heteroclinic networks in ${\mathbb{R}}^4$
• Pseudo-simple heteroclinic cycles in $R^4$
• Attractors in complex networks
• Analysis of dynamics near heteroclinic networks in ${\mathbb{R}}^4$ with a projected map