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Research Papers

# Nonlinear Dynamic Analysis of a Simplest Fractional-Order Delayed Memristive Chaotic System

[+] Author and Article Information
Wei Hu

School of Electronics and
Information Engineering,
Anhui University,
Hefei 230601, China
e-mail: hwei@ahu.edu.cn

Dawei Ding

School of Electronics and
Information Engineering,
Anhui University,
Hefei 230601, China
e-mail: dwding@ahu.edu.cn

Nian Wang

School of Electronics and
Information Engineering,
Anhui University,
Hefei 230601, China
e-mail: wn_xlb@ahu.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 24, 2016; final manuscript received November 24, 2016; published online xx xx, xxxx. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 12(4), 041003 (Jan 19, 2017) (8 pages) Paper No: CND-16-1402; doi: 10.1115/1.4035412 History: Received August 24, 2016; Revised November 24, 2016

## Abstract

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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## Figures

Fig. 1

Circuit model of the delayed memristive chaotic system

Fig. 9

Largest Lyapunov exponent of q=0.9

Fig. 10

Bifurcation diagram of τ=1.4

Fig. 11

Largest Lyapunov exponent of τ=1.4

Fig. 2

The corresponding bifurcation point τ0 in the system (6) for different fractional-order q∈[0.2,1]

Fig. 3

Bifurcation diagram: (a) q=0.6 and (b) q=0.8

Fig. 4

Time-domain diagram and phase diagram of the fractional system (6) with τ1=0.335<τ0=0.3425 and τ1=0.35>τ0=0.3425 in the initial value (0.2,−0.1) and (1,0.5)

Fig. 5

Time-domain diagram and phase diagram of the fractional system (6) with τ1=0.495<τ0=0.5004 and τ1=0.505>τ0=0.5004 in the initial value (1,1) and (1.5,1.5)

Fig. 6

The dynamical behaviors of the fractional system (6) with different τ for q=0.9: (a) one cycle (τ=1), (b) two cycles (τ=1.24), (c) four cycles (τ=1.4), and (d) chaos (τ=1.26)

Fig. 7

The detailed dynamical behaviors of the system (6) with different q for τ=1.4: (a) one cycle (q=0.8), (b) two cycles (q=0.84), (c) chaos (q=0.87), (d) chaos (q=0.87), (e) four cycles (q=0.89), (f) eight cycles (q=0.895), (g) chaos (q=0.91), (h) chaos (q=0.91), and (i) chaos (q=0.97)

Fig. 8

Bifurcation diagram of q=0.9

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