Stability of heteroclinic cycles in ring graphs
Claire M Postlethwaite1, Rob Sturman2
1Department of Mathematics, University of Auckland, Auckland 1142, New Zealand.
Chaos (Woodbury, N.Y.)
|July 1, 2022
Summary
Network structures can create robust, stable heteroclinic cycles, leading to non-ergodic dynamics. This study shows such cycles exist in ring graphs, suggesting non-ergodic behavior is typical in complex networks.
Area of Science:
- Network science
- Dynamical systems theory
- Graph theory
Background:
- Networks are ubiquitous in science, connecting interacting nodes.
- Network topology can induce invariant subspaces, unlike generic dynamical systems.
- These subspaces can stabilize otherwise unstable heteroclinic cycles.
Purpose of the Study:
- To investigate the existence and stability of heteroclinic cycles in ring graphs.
- To analyze the resulting dynamical behavior, specifically ergodicity.
- To explore the implications for complex network dynamics.
Main Methods:
- Analysis of ring graphs with nearest-neighbor or nearest-m-neighbor coupling.
- Examination of phase space dynamics to identify heteroclinic cycles.
- Investigation of the stability and attractivity of identified cycles.
Main Results:
- Identified classes of heteroclinic cycles in the phase space of ring graph dynamics.
- Demonstrated the existence of at least one asymptotically stable heteroclinic cycle.
- Showed that network-induced cycles lead to non-ergodic dynamics.
Conclusions:
- Ring graphs exhibit asymptotically stable heteroclinic cycles.
- The dynamics in these networks are non-ergodic due to stable cycles.
- Non-ergodic behavior is conjectured to be typical in less structured networks.
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