Numerical methods for stochastic differential equations.

Numerical methods for stochastic differential equations, including Taylor expansion approximations, Runge-Kutta like methods and implicit methods, are summarized. Important differences between simulation techniques with respect to the strong (pathwise) and the weak (distributional) approximation criteria are discussed. Applications to the visualization of nonlinear stochastic dynamics. the computation of Lyapunov exponents and stochastic bifurcations are also presented

A detectability criterion and data assimilation for non-linear differential equations

In this paper we propose a new sequential data assimilation method for non-linear ordinary differential equations with compact state space. The method is designed so that the Lyapunov exponents of the corresponding estimation error dynamics are negative, i.e. the estimation error decays exponentially fast. The latter is shown to be the case for generic regular flow maps if and only if the observation matrix H satisfies detectability conditions: the rank of H must be at least as great as the number of nonnegative Lyapunov exponents of the underlying attractor. Numerical experiments illustrate the exponential convergence of the method and the sharpness of the theory for the case of Lorenz96 and Burgers equations with incomplete and noisy observations

Lyapunov Exponents of Two Stochastic Lorenz 63 Systems

Two different types of perturbations of the Lorenz 63 dynamical system for Rayleigh-Benard convection by multiplicative noise -- called stochastic advection by Lie transport (SALT) noise and fluctuation-dissipation (FD) noise -- are found to produce qualitatively different effects, possibly because the total phase-space volume contraction rates are different. In the process of making this comparison between effects of SALT and FD noise on the Lorenz 63 system, a stochastic version of a robust deterministic numerical algorithm for obtaining the individual numerical Lyapunov exponents was developed. With this stochastic version of the algorithm, the value of the sum of the Lyapunov exponents for the FD noise was found to differ significantly from the value of the deterministic Lorenz 63 system, whereas the SALT noise preserves the Lorenz 63 value with high accuracy. The Lagrangian averaged version of the SALT equations (LA SALT) is found to yield a closed deterministic subsystem for the expected solutions which is found to be isomorphic to the original Lorenz 63 dynamical system. The solutions of the closed chaotic subsystem, in turn, drive a linear stochastic system for the fluctuations of the LA SALT solutions around their expected values.Comment: 19 pages, 4 figures, comments always welcome

Reduced-order Description of Transient Instabilities and Computation of Finite-Time Lyapunov Exponents

High-dimensional chaotic dynamical systems can exhibit strongly transient features. These are often associated with instabilities that have finite-time duration. Because of the finite-time character of these transient events, their detection through infinite-time methods, e.g. long term averages, Lyapunov exponents or information about the statistical steady-state, is not possible. Here we utilize a recently developed framework, the Optimally Time-Dependent (OTD) modes, to extract a time-dependent subspace that spans the modes associated with transient features associated with finite-time instabilities. As the main result, we prove that the OTD modes, under appropriate conditions, converge exponentially fast to the eigendirections of the Cauchy--Green tensor associated with the most intense finite-time instabilities. Based on this observation, we develop a reduced-order method for the computation of finite-time Lyapunov exponents (FTLE) and vectors. In high-dimensional systems, the computational cost of the reduced-order method is orders of magnitude lower than the full FTLE computation. We demonstrate the validity of the theoretical findings on two numerical examples

Phase diagram of the random frequency oscillator: The case of Ornstein-Uhlenbeck noise

We study the stability of a stochastic oscillator whose frequency is a random process with finite time memory represented by an Ornstein-Uhlenbeck noise. This system undergoes a noise-induced bifurcation when the amplitude of the noise grows. The critical curve, that separates the absorbing phase from an extended non-equilibrium steady state, corresponds to the vanishing of the Lyapunov exponent that measures the asymptotic logarithmic growth rate of the energy. We derive various expressions for this Lyapunov exponent by using different approximation schemes. This allows us to determine quantitatively the phase diagram of the random parametric oscillator.Comment: to appear in Physica

Optimal Piecewise-Linear Approximation of the Quadratic Chaotic Dynamics

This paper shows the influence of piecewise-linear approximation on the global dynamics associated with autonomous third-order dynamical systems with the quadratic vector fields. The novel method for optimal nonlinear function approximation preserving the system behavior is proposed and experimentally verified. This approach is based on the calculation of the state attractor metric dimension inside a stochastic optimization routine. The approximated systems are compared to the original by means of the numerical integration. Real electronic circuits representing individual dynamical systems are derived using classical as well as integrator-based synthesis and verified by time-domain analysis in Orcad Pspice simulator. The universality of the proposed method is briefly discussed, especially from the viewpoint of the higher-order dynamical systems. Future topics and perspectives are also provide