Richard F. Bass, Krzysztof Burdzy
We study an ordinary differential equation controlled by a stochastic process. We present results on existence and uniqueness of solutions, on associated local times (Trotter and Ray-Knight theorems), and on time and direction of bifurcation. A relationship with Lipschitz approximations to Brownian paths is also discussed.
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We are concerned with random ordinary differential equations (RODEs). Our main question of interest is how uncertainties in system parameters propagate through the possibly highly nonlinear dynamical system and affect the system's bifurcation behavior. We come up with a methodology to determine the probability of the occurrence of different types of bifurcations (sub- vs super-critical) along a given bifurcation curve based on the probability distribution of the input parameters. In a first step, we reduce the system's behavior to the dynamics on its center manifold. We thereby still capture t…
This article refines the classical notion of a stochastic D-bifurcation to the respective family of n-point motions for homogeneous Markovian stochastic semiflows, such as stochastic Brownian flows of homeomorphisms, and their generalizations. This notion essentially detects at which level $k\leq n$ the support of the invariant measure of the k-point bifurcation has more than one connected component. Stochastic Brownian flows and their invariant measures which were shown by Kunita (1990) to be rigid, in the sense of being uniquely determined by the $1$-and $2$-point motions, and hence only sto…
In this paper, we introduce concepts of pathwise random almost periodic and almost automorphic solutions for dynamical systems generated by non-autonomous stochastic equations. These solutions are pathwise stochastic analogues of deterministic dynamical systems. The existence and bifurcation of random periodic (random almost periodic, random almost automorphic) solutions have been established for a one-dimensional stochastic equation with multiplicative noise.
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, …
Mean-square Stability and Bifurcations for Dissipative SDEs
研究耗散系统在随机扰动下的动力学，重点分析均方稳定性，推导线性化系统条件，探讨非线性与线性系统均方动力学关系，提供确定性方法评估随机扰动对确定性系统平衡点稳定性的影响，并通过实例展示分岔图的生成。
Stochastic bifurcation of a three-dimensional stochastic Kolmogorov system
本文研究三维随机 Kolmogorov 系统的随机分岔，依据噪声阈值刻画唯一遍历性、三类平稳测度及全局随机周期解，并证明小噪声极限。
In this paper, we study the existence, stability and bifurcation of random complete and periodic solutions for stochastic parabolic equations with multiplicative noise. We first prove the existence and uniqueness of tempered random attractors for the stochastic equations and characterize the structures of the attractors by random complete solutions. We then examine the existence and stability of random complete quasi-solutions and establish the relations of these solutions and the structures of tempered attractors. When the stochastic equations are incorporated with periodic forcing, we obtain…
In this paper, we show that the integration of a stochastic differential equations driven by G-Brownian motion in R can be reduced to the integration of an ordinary differential equations parametrized by a variable in (Ω,F). We study the sample solutions of G-SDEs by an extention of G-Itô formula. And then we also get a comparison theorem for G-SDEs and its applications.
The bifurcation theory of ordinary differential equations (ODEs), and its application to deterministic population models, are by now well established. In this article, we begin to develop a complementary theory for diffusion-like perturbations of dynamical systems, with the goal of understanding the space and time scales of fluctuations near bifurcation points of the underlying deterministic system. To do so we describe the limit processes that arise in the vicinity of the bifurcation point. In the present article we focus on the one-dimensional case.
Nonlinear dynamical systems are sometimes under the influence of random fluctuations. It is desirable to examine possible bifurcations for stochastic dynamical systems when a parameter varies. A computational analysis is conducted to investigate bifurcations of a simple dynamical system under non-Gaussian α-stable Levy motions, by examining the changes in stationary probability density functions for the solution orbits of this stochastic system. The stationary probability density functions are obtained by numerically solving a non local Fokker-Planck equation. This allows numerically investiga…
Bifurcation theory for SPDEs: finite-time Lyapunov exponents and amplitude equations
该研究针对接近叉式分岔的 SPDE，利用有限时间 Lyapunov 指数刻画可检测分岔的参数与噪声区域，并通过约化为有限维随机微分方程分析邻近分岔的动力学。
Stochastic bifurcation in economic growth model driven by Lévy noise
本文通过引入莱维噪声改进经济成长模型，研究随机波动对经济增长的影响，揭示跳跃扩散过程在长期GDP波动中的作用，并通过数值模拟展示随机噪声导致经济分岔的机制。
Some properties of random Conley index are obtained and then a sufficient condition for the existence of abstract bifurcation points for both discrete-time and continuous-time random dynamical systems is presented. This stochastic bifurcation phenomenon is demonstrated by a few examples.
On the pitchfork bifurcation for the Chafee-Infante equation with additive noise
研究了被无限维Wiener过程扰动的Chafee-Infante方程的叉式分岔，发现尽管噪声“破坏”了分岔，但分岔残余以正的有限时间Lyapunov指数形式持续存在，并证明了体积的有限时间扩张随维度增加而增加。
We consider the averaging principle for deterministic or stochastic systems with a fast stochastic component (family of continuous-time Markov chains depending on the state of the system as a parameter). We show that, due to bifurcations in the simplex of invariant probability measures of the chains, the limiting system should be considered on a graph or on an open book with certain gluing conditions in the vertices of the graph (or on the bifurcation surface).
Long time fluctuations at critical parameter of Hopf's bifurcation
本文研究受乘性布朗运动扰动的超临界Hopf分岔系统在临界参数处的长时间涨落，通过时空尺度变换将系统近似为中心流形上的二维过程，并利用对快速角向变量取平均得到描述临界涨落的普适随机微分方程。
On the stochastic bifurcations regarding random iterations of polynomials of the form $z^{2} + c_{n}$
本文研究了多项式形式$z^{2} + c_{n}$随机迭代的随机分岔，揭示了随机Julia集的连通性与分岔半径的关系，并证明了在特定参数下Julia集的不连通性。
General stochastic Euler schemes for ordinary differential equations are studied. We give proofs on the consistency, the rate of convergence and the asymptotic normality of these procedures.
This paper is devoted to the stochastic optimal control problem of ordinary differential equations allowing for both path-dependence and measurable randomness. As opposed to the deterministic path-dependent cases, the value function turns out to be a random field on the path spaces and it is characterized by a stochastic path-dependent Hamilton-Jacobi (SPHJ) equation. A notion of viscosity solution is proposed and the value function is proved to be the unique viscosity solution to the associated SPHJ equation.
Stability analysis of the nonlinear pendulums under stochastic perturbations
本文研究了悬挂点经历随机振动的非线性摆的稳定性，利用遍历性简化随机振动平均，完整分析了平均Hamilton系统的相图分岔，并通过Poincaré返回映射揭示了其与精确随机扰动系统的对应关系。
We consider the impact of additive Gaussian white noise on a supercritical pitchfork bifurcation in an unbounded domain. As an example we focus on the stochastic Swift-Hohenberg equation with polynomial nonlinearity. Here we identify the order where small noise first impacts the bifurcation. Using an approximation via modulation equations, we provide a tool to analyse how the noise influences the dynamics close to a change of stability.
The Geometry of Quasi-Cycles: How Stoichiometric Covariance Alters Pre-Bifurcation Signatures
本文研究了在罗森茨威格-马卡罗模型中，机械推导的种群噪声如何影响近霍普夫动力学，通过比较两种具有相同确定性漂移但捕食诱导协方差结构不同的扩散闭合方法。
In contrast to existing works on stochastic averaging on finite intervals, we establish an averaging principle on the whole real axis, i.e. the so-called second Bogolyubov theorem, for semilinear stochastic ordinary differential equations in Hilbert space with Poisson stable (in particular, periodic, quasi-periodic, almost periodic, almost automorphic etc) coefficients. Under some appropriate conditions we prove that there exists a unique recurrent solution to the original equation, which possesses the same recurrence property as the coefficients, in a small neighborhood of the stationary solu…
Critical transitions (or tipping points) are drastic sudden changes observed in many dynamical systems. Large classes of critical transitions are associated to systems, which drift slowly towards a bifurcation point. In the context of stochastic ordinary differential equations (SODEs), there are results on growth of variance and autocorrelation before a transition, which can be used as possible warning signs in applications. A similar theory has recently been developed in the simplest setting also for stochastic partial differential equations (SPDEs) for self-adjoint operators in the drift ter…
Use of Topological Data Analysis for the Detection of Phenomenological Bifurcations in Stochastic Epidemiological Models
本研究针对随机流行病学模型，引入三种接触率噪声，用拓扑数据分析的同调分岔图分析SIS、SIR模型的长时间分布，发现噪声类型显著影响模型预测结果。
Impact of spinning on the early-warning signs in non-Markovian stochastic systems
研究非马尔可夫系统临界转变的早期预警信号，分析多种随机强迫，表明指标有效性受系统特性影响，比较赫斯特指数和分岔类型影响，证明旋转动力学改变标度律，提供实用指南并验证。
Warning signs for tipping points (or critical transitions) have been very actively studied. Although the theory has been applied successfully in models and in experiments for many complex systems such as for tipping in climate systems, there are ongoing debates, when warning signs can be extracted from data. In this work, we shed light on this debate by considering different types of underlying noise. Thereby, we significantly advance the general theory of warning signs for nonlinear stochastic dynamics. A key scenario deals with stochastic systems approaching a bifurcation point dynamically u…
Weighted $L_\infty$ Asymptotic Characterisation of Perturbed Autonomous Linear Ordinary and Stochastic Differential Equations: Part I -- ODEs
本文研究加性扰动下自治线性常微分方程解的渐近增长与衰减，建立强迫函数时间平均渐近行为与解速率之间的充要条件，并概述向次主导扰动情形的扩展。
On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients
本文提出了一种随机微分方程的扰动理论，用于估计解过程与任意伊藤过程之间的L^p距离，并应用该理论研究了非全局单调系数的随机常微分方程和随机偏微分方程的数值解的强收敛速率。
A stochastic perturbation approach to nonlinear bifurcating problems
本文提出利用多项式混沌展开方法，通过随机扰动检测参数系统的分支，无需先验信息并提高计算效率。
An adaptive network model using SIS epidemic propagation with link-type dependent link activation and deletion is considered. Bifurcation analysis of the pairwise ODE approximation and the network-based stochastic simulation is carried out, showing that three typical behaviours may occur; namely, oscillations can be observed besides disease-free or endemic steady states. The oscillatory behaviour in the stochastic simulations is studied using Fourier analysis, as well as through analysing the exact master equations of the stochastic model. A compact pairwise approximation for the dynamic netwo…
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