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This paper proposed a novel fractional-order memristor-based chaotic circuit. A memristive diode bridge cascaded with a fractional-order RL filter constitutes the generalized fractional-order memristor. The mathematical model of the proposed fractional-order chaotic circuit is established by extending the nonlinear capacitor and inductor in the memristive chaotic circuit to the fractional order. Detailed theoretical analysis and numerical simulations are carried out on the dynamic behavior of the proposed circuit by investigating the stability of equilibrium points and the influence of circuit parameters on bifurcations. The results show that the order of the fractional-order circuit has a great influence on the dynamical behavior of the system. The system may exhibit complicated nonlinear dynamic behavior such as bifurcation and chaos with the change of the order. The equivalent circuits of the fractional-order inductor and capacitor are also given in the paper, and the parameters of the equivalent circuits are solved by an undetermined coefficient method. Circuit simulations of the equivalent fractional-order memristive chaotic circuit are carried out in order to validate the correctness of numerical simulations and the practicability of using the integer-order equivalent circuit to substitute the fractional-order element.

Since memristor was first postulated by Leon O. Chua in 1971 [

The fingerprint of the memristor is that the loci in the voltage-current plane exhibit a pinched hysteresis loop which always passes through the origin when driven by any periodic input voltage source or current source [

Due to the nonlinear characteristics of the memristor, chaotic oscillations can easily emerge. Replacing the Chua’s diode in the canonical Chua’s oscillator with a memristor, a memristor-based chaotic circuit was implemented in 2008 [

On the other hand, fractional calculus, a more than 300-year-old mathematical subject, can describe a real object more accurately than the classical “integer” methods [

This paper is organized as follows. Section

Fractional calculus, as a generalization of integration and differentiation to integer order, can more accurately characterize the properties of actual objects. The fundamental integral-differential operator

There are three equivalent definitions used for the general fractional differentiation known as the Grünwald-Letnikov (GL) definition, the Riemann-Liouville (RL) definition, and the Caputo definition. The Caputo definition is more convenient for initial condition problems because its physical meaning is clear [

In this case, the three well-known definitions of fractional derivatives are equivalent [

Paper [

Generalized memristor and its circuit realization: (a) memristive diode bridge with series RL filter and (b) memristor symbol representation.

As nonlinear elements, inductors and capacitors can be extended to fractional order from the point of view of fractional calculus [

In order to verify whether the proposed fractional-order generalized memristor satisfied the three fingerprints of a memristor [

The hysteresis loops of the fractional-order memristor with different sinusoidal input voltages: (a)

After the realization of the fractional-order generalized memristor, we consider applying the proposed first-order generalized memristor to a classical Chua’s circuit. By replacing the traditional Chua’s oscillator with a fractional-order inductor-based memristor, a fractional-order memristive chaotic circuit is established, which is indicated in Figure

Fractional-order memristive Chua’s circuit.

The model of the fractional-order memristive Chua’s circuit is established in MATLAB, and simulations are carried out in numerical. Set capacitance

Phase diagrams of the fractional-order memristive Chua’s circuit. Phase diagrams in the (a)

In this section, equilibrium points and eigenvalues of the corresponding Jacobian matrix are calculated to qualitatively analyze the dynamic behavior of the proposed fractional-order memristive chaotic circuit. The equilibrium points of such a circuit can be obtained by solving the following equations:

Obviously,

Utilizing (

Two functions and their intersection points.

The Jacobian matrix of the characteristic (

Then the eigenvalues at equilibrium points

It can be seen that equilibrium point

The fractional-order models can increase the flexibility and degrees of freedom by means of the fractional parameters. In this section, the change of the fractional order

Bifurcation diagram of the fractional-order memristive chaotic circuit with the different order.

The phase diagrams of the fractional-order memristive Chua’s circuit with the different order.

The bifurcation diagram of

Bifurcation diagrams of the fractional-order memristive chaotic circuit and integer-order memristive chaotic circuit with different inductance

The orbits painted in red indicate the integer-order system with order

Phase diagrams of the integer-order system and fractional-order system: (a)

In this section, equivalent circuits are used to simulate the fractional-order memristive chaotic circuit. Since the inductor and capacitor can be extended to fractional order, the Oustaloup filter algorithm is used to obtain the approximate transfer function of fractional Laplace transform operator

The fractional Laplace transform operator of inductor in fractional-order generalized memristor can be physically realized using the RL chain circuits in Figure ^{−1} Hz to 10^{5} Hz.

Equivalent chain circuit of fractional-order inductor.

The resistance parameters of the equivalent chain circuit of fractional-order inductor.

0.95 | 964.42 | 12.14 | ||||||

0.98 | 30.39 |

The inductance parameters of the equivalent chain circuit of fractional-order inductor.

0.95 | 0.6896 | 0.8595 | 1.0820 | 1.3621 | 1.7147 | 0.4440 | |

0.98 | 2.4385 | 2.6473 | 2.9025 | 3.1825 | 3.4893 | 0.3367 |

Bode diagram of Oustaloup approximation and fractance circuit approximation.

After realizing the equivalent circuit of fractional-order inductor, we consider using the same method to achieve the equivalent circuit of the fractional-order capacitor. The approximate equivalence of the fractional-order capacitor can be actualized utilizing RC ladder topologies, as shown in Figure

Equivalent ladder circuit of the fractional-order capacitor.

In consideration of capacitors ^{−2} Hz to 10^{5} Hz. The chain equivalent parameters of fractional-order inductor ^{−2} Hz to 10^{5} Hz, as shown in Figure

The resistance parameters of the equivalent ladder circuit of the fractional-order capacitor.

10 | 13.80 | 109.66 | ||||||

100 | 1.38 | 10.97 |

The capacitance parameters of the equivalent ladder circuit of the fractional-order capacitor.

10 | |||||||

100 | 1.04 | 1.14 | 1.25 |

Bode diagram of Oustaloup approximation and fractance circuit approximation.

The resistance parameters of the equivalent chain circuit of fractional-order inductor.

118.80 | 1.30 |

The inductance parameters of the equivalent chain circuit of fractional-order inductor.

0.1045 | 0.1135 | 0.1244 | 0.1364 | 0.1495 | 0.0144 |

Bode diagram of Oustaloup approximation and fractance circuit approximation.

The equivalent realization of the fractional-order memristor-based chaotic circuit.

In this section, we consider utilizing PSpice to simulate the fractional-order memristive equivalent circuit. Considering order

Phase diagrams of the realized fractional-order memristive Chua’s circuit. Phase diagrams in the (a)

In this paper, a fractional-order memristor-based chaotic circuit is presented. The inductor in generalized memristor and the capacitors and inductors in the memristive chaotic circuit are all fractional order. Firstly, the mathematical model of fractional-order inductor-based generalized memristor is established, and then the characteristic equations of the fractional-order memristive chaotic circuit are derived. After that, theoretical analysis and numerical simulations are carried out such as the analysis of equilibrium point and stability and the influence of the order on the stability of the system. The results show that the fractional-order circuit exhibits different dynamic behaviors such as bifurcation and chaos with the change of the order, which indicates the importance of order effects on the dynamic behaviors of the system. In order to verify the aforementioned analysis, the equivalent circuit of the fractional-order memristive chaotic circuit is presented. The nonlinear elements inductor and capacitor are approximately equivalent through a unit circuit. Using the method of undetermined coefficients to solve circuit parameters, a fractional-order memristor-based equivalent circuit is constructed in PSpice. The results of circuit simulations well confirm the investigation on the fractional-order memristive chaotic circuit both in theoretical and numerical.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there is no conflict of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (Grant No. 51507134), Scientific Research Program Funded by Shaanxi Provincial Department of Water Resources (Grant No. 2017slkj-15), Natural Science Foundation of Shaanxi Province (Grant No. 2018JM5068), Key Project of Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2018ZDXM-GY-169), and Xi’an Science and Technology Bureau innovation project (Grant No. 201805037 (21)).