A simplest fractional-order delayed memristive chaotic system is investigated in order to analyze the nonlinear dynamics of the system. The stability and bifurcation behaviors of this system are initially investigated, where time delay is selected as the bifurcation parameter. Some explicit conditions for describing the stability interval and the transversality condition of the emergence for Hopf bifurcation are derived. The period doubling route to chaos behaviors of such a system is discussed by using a bifurcation diagram, a phase diagram, a time-domain diagram, and the largest Lyapunov exponents (LLEs) diagram. Specifically, we study the influence of time delay on the chaotic behavior, and find that when time delay increases, the transitions from one cycle to two cycles, two cycles to four cycles, and four cycles to chaos are observed in this system model. Corresponding critical values of time delay are determined, showing the lowest orders for chaos in the fractional-order delayed memristive system. Finally, numerical simulations are provided to verify the correctness of theoretical analysis using the modified Adams–Bashforth–Moulton method.

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