Noise-induced phenomena in high-dimensional dynamical systems were investigated from a random dynamical systems point of view. In a class of generalized Hénon maps, which are randomly perturbed delayed logistic maps, with monotonically increasing noise levels, we observed (i) an increase in the number of positive Lyapunov exponents from 4 to 5, and the emergence of characteristic periods at the same time, and (ii) a decrease in the number of positive Lyapunov exponents from 4 to 3, and an increase in Kolmogorov--Sinai entropy at the same time. Our results imply that simple concepts of noise-induced phenomena, such as noise-induced chaos and/or noise-induced order, may not describe those analogue in high dimensional dynamical systems, owing to coexistence of noise-induced chaos and noise-induced order.
Noise-induced statistical periodicity in a class of one-dimensional maps is studied. We show the existence of statistical periodicity in a modified Lasota-Mackey map and describe the phenomenon in terms of almost cyclic sets. A transition from a stable state to a periodic state of the density depending on the noise level is observed in numerical investigations based on trajectory averages and by means of a transfer operator approach. We conclude that the statistical periodicity is the origin of the almost periodicity in noise-induced order.
One-shot prediction of noise-induced bifurcations with reservoir computing
Graduate School of Informatics, Kyoto University（京都大学情报学研究科）；College of Science and Engineering, Chubu University（中部大学理工学部）；Department of Mathematics, Kyoto University（京都大学数学系）；Faculty of Science, Hokkaido University（北海道大学理学部）；AIT Center, Sapporo City University（札幌市立大学AIT中心）；Graduate School of Information Science and Technology, The University of Tokyo（东京大学大学院信息理工学系研究科）
In this article, the dynamics and complexity of a noise induced blood flow system have been investigated. Changes in the dynamics have been recognized by measuring the periodicity over significant parameters. Chaotic as well as non-chaotic regimes have also been classified. Further, dynamical complexity has been studied by phase space based weighted entropy. Numerical results show a strong correlation between the dynamics and complexity of the noise induced system. The correlation has been confirmed by a cross-correlation analysis.
Modeling noise induced resonance in an excitable system: An alternative approach
最近，它被观察到[MD。Nurujjaman等人，Phy. Rev. E \textbf{80}，015201（R）（2009）]，在可激发系统中，可以通过在较高噪声水平下阻断噪声对系统的破坏性影响来维持相干共振中的噪声诱导相干性。这种恒定相干共振现象不能用现有的加噪声激励系统模型方程的模拟方法来解释。本文提出了一个解释噪声诱导共振现象CCR以及相干共振（CR）和随机共振（SR）的一般模型。考虑到噪声引起的共振现象的基本机制，进行了模拟：噪声只扰动系统的控制参数，激发相干振荡，采取适当的预防措施，使噪声的破坏性影响不影响系统。在这种方法中，CR已经从系统输出和噪声之间的干扰中获得，并且SR已经通过添加噪声和亚阈值信号而获得。这也解释了在CCR中观察到的相干振荡的频移与噪声水平。
Entrainment of noise-induced and limit cycle oscillators under weak noise
描述生物系统中振荡的理论模型通常是极限环振荡器，其中确定性非线性动力学给出持续的周期振荡，或者是噪声诱导振荡器，其中固定点是线性稳定的，具有复特征值，并且添加噪声给出围绕固定点的振荡，具有波动幅度。我们以研究得很好的van der Pol方程为例，研究了每一类模型在外部周期强迫下的行为。我们发现，当强迫是添加剂，噪声诱导的振荡器可以显示只有一个到一个夹带的外部频率，在相反的极限环振荡器，这是已知的夹带到任何比例。另一方面，当外部强迫是乘法时，噪声诱导振子可以表现出比一比一更少的夹带，而极限环振子表现出任何比率的夹带。噪声模糊的夹带在一般情况下，但明确的夹带区域的极限环可以确定，只要噪声不是太强。
We prove the existence of multiple noise-induced transitions in the Lasota-Mackey map, which is a class of one dimensional random dynamical system with additive noise. The result is achieved by the help of rigorous computer assisted estimates. We first approximate the stationary distribution of the random dynamical system and then compute certified error intervals for the Lyapunov exponent. We find that the sign of the Lyapunov exponent changes at least three times when increasing the noise amplitude. We also show numerical evidence that the standard non-rigorous numerical approximation by fin…
A solvable model of noise effects on globally coupled limit cycle oscillators is proposed. The oscillators are under the influence of independent and additive white Gaussian noise. The averaged motion equation of the system with infinitely coupled oscillators is derived without any approximation through an analysis based on the nonlinear Fokker--Planck equation. Chaotic synchronization associated with the appearance of macroscopic chaotic behavior is shown by investigating the changes in averaged motion with increasing noise intensity.
The motion of weakly inertial Brownian particles, transported by steady two-dimensional fluid flows, is investigated by means of asymptotic methods. We focus on the phenomenon of noise-induced separatrix crossing, which can force particles to enter or exit recirculation cells in an unpredictable manner. An analytical expression for the probability of separatrix crossing is obtained. It can be applied to a wide variety of flows, provided some elementary kinematical quantities of the fluid flow are known. It does not require to solve particle trajectories.
