Dynamical Symmetry Breaking of Infinite-Dimensional Stochastic System

The mapping relationship between the symmetry and the conserved quantity inspired researchers to seek the conserved quantity from the viewpoint of the symmetry for the dynamic systems. However, the symmetry breaking in the dynamic systems is more common than the symmetry in the engineering. Thus, as the supplement of our previous work on the symmetry breaking of infinite-dimensional deterministic dynamic systems, the dynamical symmetry breaking of infinite-dimensional stochastic systems is discussed in this paper. Following a brief review of the stochastic (multi-)symplectic for the dynamic system excited by stochastic white noise, two types of stochastic symmetry breaking factors, including the general stochastic excitation and the general stochastic parameters of the infinite-dimensional dynamic systems, are investigated in detail. We find that both the general stochastic excitation and the general stochastic parameters will not break the local multi-symplectic structure of the dynamic systems. However, the local energy conservation law will be broken by the general stochastic excitation, as well as by the stochastic parameters, which are given by the local energy dissipation in this paper. To illustrate the validity of the analytical results, the stochastic vibration of a clamped single-walled carbon nanotube is investigated and the critical condition of the appearance of chaos is obtained. The theoretical results obtained can be used to guide us to construct the structure-preserving method for the stochastic dynamic systems.