Noise-induced statistically stable oscillations in a deterministically divergent nonlinear dynamical system

Abstract

Abstract. Inspired by partial differential equation models of homogeneous convection possessing heteroclinic connections to infinity, we study a two dimensional system of ordinary differential equations whose solutions diverge exponentially for almost all initial conditions. Random perturbations of the dynamical system destabilize the divergences resulting in stochastic oscillations. Stochastic Lyapunov function methods are used to prove the existence of a statistically stationary state. A novel Monte-Carlo method is implemented to measure the extreme statistics associated with the stochastic oscillations, and a WKB analysis at low noise amplitude is carried out to corroborate the simulations.Key words. Nonlinear dynamical systems, stochastic dynamical systems, Monte-Carlo methods, low-noise asymptotic analysis.AMS subject classifications. 60H10, 60H30, 60G40, 65C05. IntroductionNoise-induced dynamical phenomena are observed in mathematical models arising from applications in physics, chemistry, biology [9], and neuroscience [8]. In some cases a dynamical system's variability arises from underlying discreteness. One such example is an individual-based predator-prey system [12] where it was demonstrated that stochastic cycles are present in the population-level models even though the continuous (infinite population) model fails to display oscillations. Low level noise may also induce oscillating behavior in problems close to a bifurcation, e.g., close to a saddle-node [18] or sniper [7] bifurcation, and in excitable systems [13]. Even though intrinsic internal or externally imposed noise is an integral component of many processes, numerical simulations of theoretically deterministic models may be sensitive to round-off errors that serve as small (albeit artificial) random perturbations.There is evidence that numerical errors can systematically impact computational simulations of some problems. Recently, Calzavarini et al.[3] studied a model of "homogeneous" Rayleigh-Bénard convection described by nonlinear partial differential equations with spatially periodic boundary conditions. That model contains exact exponentially growing solutions, i.e., heteroclinic connections to infinity, and these solutions accurately fit numerical simulations for a finite time, followed by an unexpected sudden collapse. This pattern is repeated with collapses occurring at seemingly random times even though there is no inherent randomness in the model or in the numerical method. The only systematic error present is the small round-off noise. This example inspired us to look for a simple dynamical system with the same property, i.e., a deterministic model that blows up exponentially for almost all initial conditions

Noise-induced stabilization of planar flows II

“…Here, we also provide a radially sharp lower bound on the decay rate of the invariant measure's density as stated in Theorem 5.5 of Part I [6]. This work extends and strengthens a stream of results on similar problems [1,2,4,5,8].…”

“…In Part I of this work [6], we investigated the following complex-valued dynamics (1.1) dz t = (az n+1 t + a n z n t + · · · + a 0 ) dt + σ dB t z 0 ∈ C where n ≥ 1 is an integer, a ∈ C \ {0}, a i ∈ C, σ ≥ 0, and B t = B (1) t + iB (2) t is a complex Brownian motion defined on a probability space (Ω, F, P). There, we studied how the presence of noise (σ > 0 in (1.1)) could stabilize the unstable underlying deterministic system (σ = 0 in (1.1)).…”

Noise-induced stabilization of planar flows I

“…While this is useful in constructing Lyapunov functions, it forces us to employ a generalized Itô-Tanaka formula due to Peskir [18] to estimate contributions along curves where our Lyapunov function is not C 2 . This allows us to avoid smoothing or mollifying along the boundaries which leads to a substantial reduction in the complexity of the argument when compared to previous works [3,6,10,4]. In Section 5, we state the precise results we will actually prove which, when combined with the results in Section 4, will imply the main results as stated in Section 3.…”

“…It is important to remark that the system (1.1) and other similar planar systems have been studied before [1,2,3,4,6,10,19]. In the case when n = 1 in (1.1), the asymptotic behavior of the invariant density was first studied in [6] to help extract information on the distribution of spacing between close, heavy particles transported by moderately turbulent flows.…”

Stochastic Processes and Applications

“…Such noiseinduced stabilization phenomena are studied in two dimensions in [9,28]. The analysis is based on the construction of appropriate Lyapunov functions.…”