Equivariant degree approach to random periodic solutions in second-order stochastic systems with symmetry

Shi Yu; Shi Yu

doi:10.3934/math.2026914

AIMS Mathematics

2026, Volume 11, Issue 7: 22623-22657. doi: 10.3934/math.2026914

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Research article

Equivariant degree approach to random periodic solutions in second-order stochastic systems with symmetry

Shi Yu ^,

Department of Mathematics, Odessa College, 201 W. University Blvd., Odessa, TX 79764, USA

Received: 27 May 2026 Revised: 10 July 2026 Accepted: 14 July 2026 Published: 27 July 2026
MSC : 37G40, 37H10, 47H11, 55M25

We study second-order stochastic systems with finite symmetry through an associated deterministic mean-field equation. This reduction provides a symmetry-preserving setting in which equivariant degree methods can be applied to the periodic problem. After reformulating the mean-field equation as a compact perturbation of the identity, we define a mean-field equivariant degree and introduce a degree-jump invariant for detecting bifurcation from the trivial branch. A nonzero degree jump yields nontrivial $2\pi$-periodic solutions, while its Burnside-ring components provide information on possible symmetry types of the bifurcating branches. We then formulate conditional lifting principles connecting mean-field periodic solutions with random periodic solutions of the original stochastic system through measurable complete random trajectories with prescribed mean motion. A sufficient framework based on a compact invariant random set and exponential contraction is also established. Examples with dihedral symmetry illustrate the computation of the invariant and the detection of maximal orbit types.
- random periodic solutions,
- equivariant degree,
- stochastic systems,
- symmetry,
- bifurcation
Citation: Shi Yu. Equivariant degree approach to random periodic solutions in second-order stochastic systems with symmetry[J]. AIMS Mathematics, 2026, 11(7): 22623-22657. doi: 10.3934/math.2026914

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Abstract

We study second-order stochastic systems with finite symmetry through an associated deterministic mean-field equation. This reduction provides a symmetry-preserving setting in which equivariant degree methods can be applied to the periodic problem. After reformulating the mean-field equation as a compact perturbation of the identity, we define a mean-field equivariant degree and introduce a degree-jump invariant for detecting bifurcation from the trivial branch. A nonzero degree jump yields nontrivial $2\pi$-periodic solutions, while its Burnside-ring components provide information on possible symmetry types of the bifurcating branches. We then formulate conditional lifting principles connecting mean-field periodic solutions with random periodic solutions of the original stochastic system through measurable complete random trajectories with prescribed mean motion. A sufficient framework based on a compact invariant random set and exponential contraction is also established. Examples with dihedral symmetry illustrate the computation of the invariant and the detection of maximal orbit types.

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