@article{Yu2026, 
author = {Shi Yu},
title = {Equivariant degree approach to random periodic solutions in second-order stochastic systems with symmetry},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {7},
pages = {22623-22657},
keywords = {random periodic solutions, equivariant degree, stochastic systems, symmetry, bifurcation},
url = {https://www.sciopen.com/article/10.3934/math.2026914},
doi = {10.3934/math.2026914},
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  π-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.}
}