Analysis of the equilibrium positions of nonlinear dynamical systems in the presence of coarse-graining disturbance in space

Abstract

Loosely speaking, a coarse-grained space is a space in which the generic point is not infinitely thin, but rather has a thickness; and here this feature is modelled as a space in which the generic increment is not dx, but rather (dx) α , 0 < α < 1. The purpose of the article is to analyze the non-linearity so induced by this coarse-graining effect. This approach via (dx) α leads us to the use of fractional analysis which so provides models in the form of nonlinear differential equations of fractional order. The purpose of the paper is to examine how the coarse-graining effect affects the equilibrium position of dynamical systems, and to this end, one uses the linearization technique on the one hand, and the Lyapunov function approach on the other hand in the framework of fractional analysis. It is shown that in some instances, the coarsegraining phenomenon can convert the initial stable system into an unstable system.

Multi-wave solutions of the space–time fractional Burgers and Sharma–Tasso–Olver equations

“…In this section we present the generalized exp-method to construct exact analytical solutions of nonlinear FDEs with the modified Riemann-Liouville derivative defined by Jumarie [15][16][17][18][19]. Assume that f : R !…”

“…Caputo put definitions which give zero value for fractional differentiation of constant function, but these definitions require that the function should be smooth and differentiable [11,12]. Recently, Jumarie derived definitions for the fractional integral and derivative called modified Riemann-Liouville [15][16][17][18][19], which are suitable for continuous and non-differentiable functions and give differentiation of a constant function equal to zero. The modified Riemann-Liouville fractional definitions are used effectively in many different problems [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37].…”

Axiomatic Local Metric Derivatives for Low-Level Fractionality with Mittag-Leffler Eigenfunctions

“…Some authors have been attempting to come over those difficulties by redefining some expressions of fractional integral and derivatives [9], arguing that by this way the basic rules of usual differential and integral calculus would be nonviolated. Unfortunately several mistakes here pointed out for these attempts [13,14,15,16,17].…”

“…In this way, we try to make clear, in short, that existing studies in the literature that apply the Modified Riemann Liouville (MRL) derivative definition [9], and there are so many, are in fact local and behave as an approximations but, provided that the order of the derivative is very close to one. That is, being local and considered as approximations.…”

Fractional thermoelectric viscoelastic materials

“…In this study, the new fractional Taylor's series of time‐fractional order α developed by Jumarie57 was adopted to expand q ( x,t + τ 0 ), and with terms up to α in the thermal τ 0 retained, we obtained …”