Stability of Fixed Points and Chaos in Fractional Systems

TL;DR

This paper introduces a method to determine the stability range of fixed points in discrete fractional systems and explores the implications for chaos in continuous fractional systems.

Contribution

It proposes a novel method for stability analysis in fractional systems and conjectures the absence of chaos in their continuous counterparts.

Findings

The method effectively determines fixed point stability ranges.

Chaos is conjectured to be impossible in continuous fractional systems.

The approach is tested on fractional generalizations of standard and logistic maps.

Abstract

In this paper we propose a method to define the range of stability of fixed points for a variety of discrete fractional systems of the order $0 < \alpha <2$. 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.