Complete Heteroclinic Networks Derived from Graphs Consisting of Two Cycles
Abstract
Abstract We address the question how a given connection structure (directed graph) can be realised as a heteroclinic network that is complete in the sense that it contains all unstable manifolds of its equilibria. For a directed graph consisting of two cycles, we provide a constructive method to achieve this: (i) enlarge the graph by adding some edges and (ii) apply the simplex method to obtain a network in phase space. Depending on the length of the cycles, we derive the minimal number of required new edges. In the resulting network, each added edge leads to a positive transverse eigenvalue at the respective equilibrium. We discuss the total number of such positive eigenvalues in an individual cycle and some implications for the stability of this cycle.
Fields of Study & Subjects
Graph theory and applications
Advanced Graph Theory Research
Interconnection Networks and Systems
Publication details
- Journal / source
- Journal of Nonlinear Science
- Publisher
- Springer Science and Business Media LLC
- Volume
- 36
- Issue
- 3
- Publication date
- 18 April 2026
- Language
- EN