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 $D^{\alpha }x\left(t\right)=\delta x(t-\tau )-ϵx{(t-\tau )}^3-px{\left(t\right)}^2+qx\left(t\right).$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 $\delta$, $ϵ$, $p$, $q$ 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 $ϵ$ 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.

分数导数和延迟是对自然系统中的记忆特性进行建模的重要工具。这项工作涉及分数阶延迟微分方程的稳定性分析$D^{\alpha }X\left(吨\right)=\delta X(吨-\tau )-\epsilon X{(吨-\tau )}^3-pX{\left(吨\right)}^2+qX\left(吨\right).$我们在平衡点附近提供该系统的线性化，并提出线性化稳定性条件。为了讨论平衡点的稳定性，我们提出了关于参数的各种条件$\delta$,$\epsilon$,$p$,$q$和$\tau$. 即使系统中涉及五个参数，我们也能够在$q\delta -$平面为任何积极$\epsilon$和$p$. 这提供了系统稳定性的完整分析。此外，我们研究了所提出模型中的混沌。该系统在很宽的延迟参数范围内表现出混沌。