On stability of fixed points and chaos in fractional systems.

Abstract

In this paper, we propose a method to calculate asymptotically period two sinks and define the range of stability of fixed points for a variety of discrete fractional systems of the order 0<α<2 . The method is tested on various forms of fractional generalizations of the standard and logistic maps. Based on our analysis, we make a conjecture that chaos is impossible in the corresponding continuous fractional systems. Many natural (biological, physical, etc.) and social systems possess power-law memory and can be described by the fractional differential/difference equations. Nonlinearity is an important property of these systems. Behavior of such systems can be very different from the behavior of the corresponding systems with no memory. Previous research on the issues of the first bifurcations and the stability of fractional systems mostly addressed the question of sufficient conditions. In this paper, we propose the equations that allow calculations of the coordinates of the asymptotically stable period two sinks and the values of nonlinearity and memory parameters defining the first bifurcation form the stable fixed points to the T = 2 sinks.