Dynamical Analysis Of Fractional Order Generalized Logistic Map

TL;DR

This paper introduces a fractional order generalized logistic map with richer dynamics, analyzes its stability and bifurcations, and explores chaos control and synchronization methods.

Contribution

It extends the classical logistic map by incorporating fractional order and an additional parameter, providing new insights into its stability, bifurcation, and control.

Findings

The generalized map exhibits complex bifurcation structures.

Chaos can be effectively controlled using delayed feedback.

The system shows multistability and synchronization capabilities.

Abstract

In this work, we propose a generalization to the classical logistic map. The generalized map preserves most properties of the classical map and has richer dynamics as it contains the fractional order and one more parameter. We propose the stability bounds for each equilibrium point. The detailed bifurcation analysis with respect to both parameters is presented using the bifurcation diagrams in one and two dimensions. The chaos in this system is controlled using delayed feedback. We provide some non-linear feedback controllers to synchronize the system. The multistability in the proposed system is also discussed.