Stability and Bifurcation Analysis of a Fractional Order Delay Differential Equation Involving Cubic Nonlinearity

TL;DR

This paper analyzes the stability and bifurcation behavior of a fractional order delay differential equation with cubic nonlinearity, providing stability conditions, parameter space sketches, and chaos investigation.

Contribution

It offers a comprehensive stability analysis of a fractional delay differential equation with cubic nonlinearity, including stability conditions and chaos exploration.

Findings

Stable regions in the parameter space are identified.

The system exhibits chaos for a wide range of delay values.

Linearization and stability conditions are established for equilibrium points.

Abstract

Fractional derivative and delay are important tools in modeling memory properties in the natural system. This work deals with the stability analysis of a fractional order delay differential equation \begin{equation*} D^\alpha x(t)=\delta x(t-\tau)-\epsilon x(t-\tau)^3-px(t)^2+q x(t). \end{equation*} We provide linearization of this system in a neighbourhood of equilibrium points and propose linearized stability conditions. To discuss the stability of equilibrium points, we propose various conditions on the parameters $\delta$, $\epsilon$, $p$, $q$ and $\tau$. Even though there are five parameters involved in the system, we are able to provide the stable region sketch in the $q\delta-$plane for any positive $\epsilon$ and $p$. This provides the complete analysis of stability of the system. Further, we investigate chaos in the proposed model. This system exhibits chaos for a wide range of delay parameter.