Hopf bifurcation with additive noise
Hopf bifurcation with additive noise
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DOI：
10.1088/1361-6544/aad208
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发表时间：
2017-10
期刊：
影响因子：
1.7
通讯作者：
T. S. Doan;Maximilian Engel;J. Lamb;M. Rasmussen
中科院分区：
文献类型：
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作者：
T. S. Doan;Maximilian Engel;J. Lamb;M. Rasmussen
我们考虑了加性白色噪声作用下的二维常微分方程的动力学行为，并确定了三个动力学相：（I）具有一致同步轨迹的随机吸引子，（II）具有非一致同步轨迹的随机吸引子和（III）不具有同步轨迹的随机吸引子.在阶段（I）和（II）中的随机吸引子是具有负的李雅普诺夫指数的随机平衡点，而在阶段（III）中存在具有正的李雅普诺夫指数的所谓随机奇怪吸引子。我们分析了不同的动力学阶段的发生作为一个函数的线性稳定性的起源（确定性的Hopf分岔参数）和剪切（振幅相位耦合参数）。我们表明，小剪切意味着同步，并获得同步不能均匀的线性稳定性的情况下，在原点或存在足够强的剪切。我们提供的数值结果支持的猜想，无论线性稳定的起源，有一个临界强度的剪切系统动力学失去同步，并进入阶段（III）。
We consider the dynamics of a two-dimensional ordinary differential equation exhibiting a Hopf bifurcation subject to additive white noise and identify three dynamical phases: (I) a random attractor with uniform synchronisation of trajectories, (II) a random attractor with non-uniform synchronisation of trajectories and (III) a random attractor without synchronisation of trajectories. The random attractors in phases (I) and (II) are random equilibrium points with negative Lyapunov exponents while in phase (III) there is a so-called random strange attractor with positive Lyapunov exponent. We analyse the occurrence of the different dynamical phases as a function of the linear stability of the origin (deterministic Hopf bifurcation parameter) and shear (amplitude-phase coupling parameter). We show that small shear implies synchronisation and obtain that synchronisation cannot be uniform in the absence of linear stability at the origin or in the presence of sufficiently strong shear. We provide numerical results in support of a conjecture that irrespective of the linear stability of the origin, there is a critical strength of the shear at which the system dynamics loses synchronisation and enters phase (III).