Abstract
We combine the stochastic perturbation method with the maximum entropy principle to construct approximations of the first probability density function of the steady-state solution of a class of nonlinear oscillators subject to small perturbations in the nonlinear term and driven by a stochastic excitation. The nonlinearity depends both upon position and velocity, and the excitation is given by a stationary Gaussian stochastic process with certain additional properties. Furthermore, we approximate higher-order moments, the variance, and the correlation functions of the solution. The theoretical findings are illustrated via some numerical experiments that confirm that our approximations are reliable.
Highlights
Introduction and MotivationThe analysis of stochastic perturbations in nonlinear dynamical systems is a hot topic in applied mathematics [1,2] with many applications in apparently different areas such as control [3], economy [4] and especially in dealing with nonlinear vibratory systems.The study of systems subject to vibrations is encountered, for example, in Physics and in Engineering
Oscillators in Physics and Engineering systems have been extensively studied in the deterministic case [5,6], and in the nonlinear case [7,8,9], due to the above-mentioned facts the stochastic analysis is more suitable since provides better understanding of their dynamics
We address the study of random cross-nonlinear oscillators subject to small perturbations affecting the nonlinear term, g, which depend on both the position, X (t), and the velocity, Ẋ (t), Ẍ (t) + 2ζω0 Ẋ (t) + eg( X (t), Ẋ (t)) + ω02 X (t) = Y (t)
Summary
The analysis of stochastic perturbations in nonlinear dynamical systems is a hot topic in applied mathematics [1,2] with many applications in apparently different areas such as control [3], economy [4] and especially in dealing with nonlinear vibratory systems. We combine mean square calculus and the stochastic perturbation method to study a class of nonlinear oscillators whose nonlinear term, g, involves both position, X (t), as velocity, Ẋ (t), we consider the case g = g( X (t), Ẋ (t)) = X 2 (t) Ẋ (t). These expressions will be given in terms of certain integrals of the correlation function of the Gaussian noise, Y (t), and of the classical impulse response function to the linearized oscillator associated to Equation (2).
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