Computationally Efficient Simulations of Stochastically Perturbed Nonlinear Dynamical Systems

Thomas Breunung and Balakumar Balachandran, Computationally Efficient Simulations of Stochastically Perturbed Nonlinear Dynamical Systems, J. Comput. Nonlinear Dynam. Sep 2022, 17(9): 091008, https://doi.org/10.1115/1.4054932

Dynamical system models can be used to describe natural and engineering systems that evolve in time. Furthermore, most natural processes are inherently nonlinear and face uncertainties stemming from, for example, parameter variations or unknown environmental conditions. Models used to describe such systems can be grouped under stochastic, nonlinear dynamical systems. Here, the authors build on numerical integration routines meant for deterministic systems and present an algorithm to compute responses of stochastic nonlinear systems. With this approach, the well-developed deterministic tools can be used in stochastic system simulations. The algorithm’s performance is demonstrated by using numerical examples, including a system with two-hundred dimensions. This algorithm can be used to compute sample paths of stochastic dynamical systems about two orders of magnitude faster compared to established numerical stochastic integration routines. In addition, a deduced Gaussian kernel enables computations of the time-varying probability density function. With this approach, one can reduce the sample size significantly and thus enable computational investigations of higher dimensional systems that are infeasible with currently available methods. The algorithm discussed here can be used as a basis for computationally efficient investigations into stochastic dynamical systems over long time spans.