Hopf bifurcation and chaos in a fractional order delayed memristor-based chaotic circuit system

Abstract

This paper present Hopf bifurcation and chaos in a fractional order delayed memristor-based chaotic circuit system. Firstly, regarding the time delay τ as a bifurcation parameter, we investigate the stability and bifurcation behaviors of this fractional order delayed memristor-based chaotic circuit system. Some explicit conditions for describing the stability interval and emergence of Hopf bifurcation are derived. Secondly, corresponding to different system parameters, the complex dynamics behaviors of this system are discussed by using the bifurcation diagrams, the Max Lyapunov exponents (MLEs) diagram, the time domain waveforms, the phase portraits and the power spectrums. Thirdly, we study the influence of the two parameters (time delay τ and fractional order q) on the chaotic behavior, and it is found when time delay τ and fractional order q increases, the transitions from period one to period two and period four to chaos are observed in this memristor–based system. Meanwhile, corresponding critical values of time delay τ and fractional order q, the lowest orders q and the minimum time delay τ for generating chaos in the fractional order delayed memristor–based system are determined, respectively. Also, when the system occurs period one, the corresponding frequency is verified theoretically and experimentally. Finally, numerical simulations are provided to demonstrate the validity of theoretical analysis using the improved Adams–Bashforth–Moulton algorithm.