Stability of heteroclinic cycles: a new approach based on a replicator equation. (English) Zbl 1542.34045
J. Nonlinear Sci. 33, No. 6, Paper No. 99, 51 p. (2023); correction ibid. 33, No. 6, Paper No. 121, 3 p. (2023).
Summary: This paper analyses the stability of cycles within a heteroclinic network formed by six cycles lying in a three-dimensional manifold, for a one-parameter model developed in the context of polymatrix replicator equations. We show the asymptotic stability of the network for a range of parameter values compatible with the existence of an interior equilibrium and we describe an asymptotic technique to decide which cycle (within the network) is visible in numerics. The technique consists of reducing the relevant dynamics to a suitable one-dimensional map, the so-called projective map. The stability of the fixed points of the projective map determines the stability of the associated cycles. The description of this new asymptotic approach is applicable to more general types of networks and is potentially useful in computational dynamics.
MSC:
| 34D20 | Stability of solutions to ordinary differential equations |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
| 37C75 | Stability theory for smooth dynamical systems |
| 92B20 | Neural networks for/in biological studies, artificial life and related topics |
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
Keywords:
asymptotic stability; polymatrix replicator; heteroclinic network; heteroclinic cycle; projective mapReferences:
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