Fractional Order Periodic Maps: Stability Analysis and Application to the Periodic-2 Limit Cycles in the Nonlinear Systems

TL;DR

This paper analyzes the stability of period-2 orbits in fractional difference equations, revealing complex conditions depending on parameters and demonstrating their application to nonlinear systems.

Contribution

It introduces stability conditions for period-2 orbits in fractional maps, extending the analysis to nonlinear systems and higher periods.

Findings

Stability depends on parameters a, b, and their sum and product.

No superstable period-2 orbits exist in fractional maps.

Conditions are validated numerically.

Abstract

We consider the stability of periodic map with period-$2$ in linear fractional difference equations where the function is $f(x)=ax$ at even times and $f(x)=bx$ at odd times. The stability of such a map for an integer order map depends on product $ab$. The conditions are much complex for fractional maps and depend on $ab$ as well as $a+b$. There are no superstable period-2 orbits. These conditions are useful in obtaining stability conditions of asymptotically periodic orbits with period-$2$ in the nonlinear case. The stability conditions are demonstrated numerically. The formalism can be generalized to higher periods.