We investigate the bifurcation phenomena for stochastic systems with multiplicative Gaussian noise, by examining qualitative changes in mean phase portraits. Starting from the Fokker–Planck equation for the probability density function of solution processes, we compute the mean orbits and mean equilibrium states. A change in the number or stability type, when a parameter varies, indicates a stochastic bifurcation. Specifically, we study stochastic bifurcation for three prototypical dynamical systems (i.e. saddle-node, transcritical, and pitchfork systems) under multiplicative Gaussian noise, and have found some interesting phenomena in contrast to the corresponding deterministic counterparts.

中文翻译：

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具有乘性高斯噪声的随机动力系统平均相位图的分岔

我们通过检查平均相位图的质变来研究具有乘法高斯噪声的随机系统的分岔现象。从求解过程的概率密度函数的 Fokker-Planck 方程开始，我们计算平均轨道和平均平衡状态。当参数变化时，数量或稳定性类型的变化表明随机分叉。具体来说，我们研究了乘性高斯噪声下三个原型动力系统（即鞍节点、跨临界和干草叉系统）的随机分岔，并发现了一些与相应确定性对应物形成对比的有趣现象。