Li Yang, Duan Jinqiao, Liu Xianbin, Zhang Yanxia
State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Nanjing 210016, China.
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, USA.
Chaos. 2020 Jun;30(6):063142. doi: 10.1063/5.0006292.
The emergence of the exit events from a bounded domain containing a stable fixed point induced by non-Gaussian Lévy fluctuations plays a pivotal role in practical physical systems. In the limit of weak noise, we develop a Hamiltonian formalism under the Lévy fluctuations with exponentially light jumps for one- and two-dimensional stochastic dynamical systems. This formalism is based on a recently proved large deviation principle for dynamical systems under non-Gaussian Lévy perturbations. We demonstrate how to compute the most probable exit path and the quasi-potential by several examples. Meanwhile, we explore the impacts of the jump measure on the quasi-potential quantitatively and on the most probable exit path qualitatively. Results show that the quasi-potential can be well estimated by an approximate analytical expression. Moreover, we discover that although the most probable exit paths are analogous to the Gaussian case for the isotropic noise, the anisotropic noise leads to significant changes in the structure of the exit paths. These findings shed light on the underlying qualitative mechanism and quantitative feature of the exit phenomenon induced by non-Gaussian noise.
由非高斯 Lévy 涨落引起的、从包含稳定不动点的有界域中出现的退出事件，在实际物理系统中起着关键作用。在弱噪声极限下，我们针对一维和二维随机动力系统，在具有指数轻跳的 Lévy 涨落下发展了一种哈密顿形式。这种形式基于最近证明的非高斯 Lévy 扰动下动力系统的大偏差原理。我们通过几个例子展示了如何计算最可能的退出路径和准势能。同时，我们定量地研究了跳跃测度对准势能的影响，定性地研究了其对最可能退出路径的影响。结果表明，准势能可以通过一个近似解析表达式得到很好的估计。此外，我们发现，尽管对于各向同性噪声，最可能的退出路径与高斯情况类似，但各向异性噪声会导致退出路径结构发生显著变化。这些发现揭示了由非高斯噪声引起的退出现象的潜在定性机制和定量特征。
