Stability and bifurcation analysis of a fractional order delay differential equation involving cubic nonlinearity

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 D(alpha)x(t) = delta x(t - tau) - epsilon x(t -tau)(3) -px(t)(2) + qx(t). We provide linearization of this system in a neighborhood 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.

Automated summary

This study focuses on the stability analysis of a fractional order delay differential equation and provides linearized stability conditions. The stable region sketch in the q delta-plane is provided for any positive epsilon and p. Additionally, chaos in the proposed model is investigated for a wide range of delay parameter.