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Filtering for a Duffing-van der Pol stochastic differential equation

Applied Mathematics and Computation
Publication Date
DOI: 10.1016/j.amc.2013.10.038
  • Continuous State-Discrete Measurement System
  • The Fokker–Planck Equation
  • Modified Second-Order Filter
  • Stochastic Differential Equations
  • Mathematics


Abstract The Stochastic Duffing-van der Pol (SDvdP) system is a non-trivial case in stochastic system theory, since it involves linear, non-linear vector fields and state-dependent stochastic accelerations. Notably, analyzing specific and non-trivial cases brings greater theoretical insights as well as reveals joining points between different topics. In the theory of stochastic differential systems, the filtered estimates are more accurate in contrast to the predicted, since the observation correction terms associated with the filtering equations contribute to the effectiveness of the estimation procedure. Thus, it becomes reasonable to analyze the stochastic Duffing-van der Pol system from the filtering viewpoint. In this paper, we wish to revisit the stochastic Duffing-van der Pol ‘filtering’ in the Fokker–Planck setting in lieu of the filtering in the Kushner setting. In the Fokker–Planck setting, observations are accounted at discrete-time instants. As a result of this, we arrive at continuous-discrete filtering equations. The Duffing-van der Pol filtering equations of this paper can be regarded as a consequence of the two-stage estimation procedure, since this paper utilizes stochastic differential equation formalism for the stochastic system of this paper. On the other hand, a discrete-time stochastic evolution of the observation is accounted for in this study. Notably, filtering equations of the paper explain explicit contributions of the predicted estimate and the observation noise to the filtered estimate. As a result of this, the filtering approach of this paper offers greater convenience for examining the filtering efficacy via simulations for specific cases. The filtering theory of this paper will be of interest to applied mathematicians, control theorists aspiring for understanding the continuous-discrete filtering better, especially for stochastic problems arising from technology, where observation rates are less.

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