Invariant Sphere Theorem and Ring-Coupled Systems
Abstract
In this work, we show how heteroclinic networks can arise in a simple class of network dynamical systems through the application of the Invariant Sphere Theorem. Ring-coupled systems are ODE networks in ℝn in which each variable xi interacts only with its predecessor xi−1. We derive conditions on the coefficients of a cubic polynomial that guarantee the existence of a globally attracting invariant sphere via the Invariant Sphere Theorem. Moreover, setting one of these coefficients to zero, we identify conditions on the remaining coefficients that guarantee the existence of a heteroclinic network on the invariant sphere. Focusing on the case n=3, we investigate perturbations of the vanishing coefficient in a neighbourhood of the heteroclinic network. Under such perturbations, the heteroclinic network is destroyed and periodic orbits emerge. When the original heteroclinic network is asymptotically stable, the resulting periodic orbits shadow the network. In one parameter regime, a unique attracting periodic orbit appears and shadows the entire heteroclinic network. By contrast, when the heteroclinic network is not stable, repelling periodic orbits arise that shadow only part of the heteroclinic structure, namely half of the heteroclinic connections. Symmetry plays a fundamental role throughout the analysis. Exploiting the symmetries of the system, we reduce the heteroclinic network to two homoclinic orbits. Furthermore, the local dynamics near the heteroclinic network can be studied on a quotient space consisting of an annulus attached to a Möbius band, with each homoclinic orbit lying on one of these components. This reduction provides a geometric framework for understanding the bifurcations and the emergence of the periodic dynamics.
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BibTeX
@misc{soares2026invariant,
title = {{Invariant Sphere Theorem and Ring-Coupled Systems}},
author = {Pedro Soares},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28223},
eprint = {2608.28223},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28223},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Invariant Sphere Theorem and Ring-Coupled Systems
AU - Pedro Soares
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28223
UR - https://arxiv.org/abs/2608.28223
AB - In this work, we show how heteroclinic networks can arise in a simple class of network dynamical systems through the application of the Invariant Sphere Theorem. Ring-coupled systems are ODE networks in ℝ^n in which each variable x_i interacts only with its predecessor x_i−1. We derive conditions on the coefficients of a cubic polynomial that guarantee the existence of a globally attracting invariant sphere via the Invariant Sphere Theorem. Moreover, setting one of these coefficients to zero, we identify conditions on the remaining coefficients that guarantee the existence of a heteroclinic network on the invariant sphere. Focusing on the case n=3, we investigate perturbations of the vanishing coefficient in a neighbourhood of the heteroclinic network. Under such perturbations, the heteroclinic network is destroyed and periodic orbits emerge. When the original heteroclinic network is asymptotically stable, the resulting periodic orbits shadow the network. In one parameter regime, a unique attracting periodic orbit appears and shadows the entire heteroclinic network. By contrast, when the heteroclinic network is not stable, repelling periodic orbits arise that shadow only part of the heteroclinic structure, namely half of the heteroclinic connections. Symmetry plays a fundamental role throughout the analysis. Exploiting the symmetries of the system, we reduce the heteroclinic network to two homoclinic orbits. Furthermore, the local dynamics near the heteroclinic network can be studied on a quotient space consisting of an annulus attached to a Möbius band, with each homoclinic orbit lying on one of these components. This reduction provides a geometric framework for understanding the bifurcations and the emergence of the periodic dynamics.
ER -