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 α x (t) = δ x (t − τ) − ϵ x (t − τ) 3 − p x (t) 2 + q x (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 δ , ϵ , p , q and τ. Even though there are five parameters involved in the system, we are able to provide the stable region sketch in the q δ − plane for any positive ϵ 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. • Delay differential equation with cubic nonlinearity with fractional order is discussed. • Stability analysis is provided by using the linearization near equilibrium points. • Stable region is sketched in the parameter plane q – δ. • Stability results are valid for any positive parameters p and ϵ. • Chaos is observed for a wide range of parameters. [ABSTRACT FROM AUTHOR]