论文标题
随机动态干草叉分叉带有添加剂Lévy噪声
The random dynamical pitchfork bifurcation with additive Lévy noises
论文作者
论文摘要
本文涉及添加剂非高斯lévy噪声对干草叉分叉的影响。我们考虑两种类型的噪声：$α$稳定的过程和截断过程。在$α$稳定的过程和截断过程中，经典的干草叉分叉模型存在着独特的不变度。与不变度度量相关的Lyapunov指数对于截断情况下的系统始终为负。随机的干草叉分叉仍发生。在这两种情况下，可吸引力均匀性，有限的Lyapunov指数和二分法谱的行为随分叉参数的变化而变化。 与布朗运动相比，莱维过程有两个关键困难。无法明确求解固定密度，因此我们必须正确估计它。这是通过强大的最大原则克服的。需要分析莱维过程的双侧上流。这是由强大的马尔可夫财产获得的。基于它们，我们建立了指示随机干草叉分叉的主要结果。
This paper concerns the effects of additive non-Gaussian Lévy noises on the pitchfork bifurcation. We consider two types of noises, $α$-stable process and the truncated process. Under both $α$-stable process and the truncated process, the classical pitchfork bifurcation model exists a unique invariant measure. The Lyapunov exponent associated with the invariant measure is always negative for the system under the truncated case. While the stochastic pitchfork bifurcation still occurs. In both cases, the attractivity uniformity, the finite-time Lyapunov exponent, and the dichotomy spectrum behave varies with the bifurcation parameter changing. Compared with Brownian motion, there is two key difficulties for the Lévy processes. The stationary density can not be solved explicitly, thus we have to estimate it properly. This is overcome by the strong maximum principle. The bilateral suprema for Lévy processes need to be analyzed. This is acquired by the strong Markov property. Based on them, we establish the main results indicating stochastic pitchfork bifurcation.
