Derivation of amplitude equations for nonlinear oscillators subject to arbitrary forcing

Abstract

By using a generalization of the multiple scales technique we develop a method to derive amplitude equations for zero-dimensional forced systems. The method allows to consider either additive or multiplicative forcing terms and can be straightforwardly applied to the case that the forcing is white noise. We give examples of the use of this method to the case of the van der Pol-Duffing oscillator. The writing of the amplitude equations in terms of a Lyapunov potential allow us to obtain an analytical expression for the probability distribution function which reproduces reasonably well the numerical simulation results.

Turing Bifurcation and Pattern Formation of Stochastic Reaction-Diffusion System

“…and , in the Appendix. Using the Fredholm solubility condition again, we can obtain And then we substitute system (28) and (33) into (24) to simplify [32]; we obtain the expressions of the coefficients of 0 , , and in the Appendix. And…”

“…It is known that amplitude equation is not only a promising tool to investigate the RDS but also the main focus of the pattern dynamics [29,30]. However, the amplitude equation is a complex process [31], and only a few systems have been chosen in the past for amplitude equation [32][33][34][35]. In this paper, we studied pattern selection of amplitude equation with a pair of wave vector by using the standard multiple scale analysis [36,37].…”

Pattern formation in the FitzHugh–Nagumo model

“…As we know that the effect of space on FN model was not taken into account in the previous works or some results were obtained by choosing particular initial conditions [27]. Especially the reaction term of fractional order is never considered and do not have a formal way to solve it [28][29][30][31].…”

Turing Instability and Amplitude Equation of Reaction-Diffusion System with Multivariable

“…Recently, the pattern formation with three or four variables has been investigated, and it obtains promising results [19,20], and Xu et al made a concrete analysis with three variables [21]. As we all know that amplitude equation is a promising tool to investigate the pattern dynamics of the reaction-diffusion system [2,22], however, the amplitude equation is a complex process [3], and the researcher often chose the amplitude equation [23][24][25][26] to investigate the reaction-diffusion system. In conclusion, spatial patterns in reaction-diffusion systems have attracted the interest of experimentalists and theorists during the last few decades.…”