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In this paper, a fractional-order chaotic circuit based on a novel fractional-order generalized memristor is proposed. It is proved that the circuit based on the diode bridge cascaded with fractional-order inductor has volt-ampere characteristics of pinched hysteresis loop. Then the mathematical model of the fractional-order memristor chaotic circuit is obtained. The impact of the order and system parameters on the dynamic behaviors of the chaotic circuit is studied by phase trajectory, Poincaré Section, and bifurcation diagram method. The order, as an important parameter, can increase the degree of freedom of the system. With the change of the order and parameters, the circuit will exhibit abundant dynamic behaviors such as coexisting upper and lower limit cycle, single scroll chaotic attractors, and double scroll chaotic attractors under different initial conditions. And the system exhibits antimonotonic behavior of antiperiodic bifurcation with the change of system parameters. The equivalent circuit simulations are designed to verify the results of the theoretical analysis and numerical simulation.

The memristor, which is considered as the fourth basic circuit element, was first proposed theoretically by Professor Leon Chua in 1971[

If the volt-ampere characteristics of the circuit ports have three characteristics fingerprints as described in [

The concept of fractional calculus is a development in the field of mathematics, which can be applied to describe memristor characteristics. Fractional calculus plays an important role in signal and image processing [

The rest of this paper mainly includes the following five sections. In Section

Fractional calculus can be regarded as an extension of classical integer-order calculus, but it has its own unique logic and grammar rules. There are several different definitions involving the fractional versions of the integral and derivative operators: Riemann-Liouville (RL) definition, Grunwald-Letnikov (GL) definition, and Caputo definition [

Because the definition of Caputo allows integer-order initial conditions to be used to solve fractional-order differential equations, it is widely used in the modeling of practical problems.

The definition of Caputo is

Laplace transform is a common tool for describing fractional-order systems. The integer-order Laplace transform can be generalized to the fractional-order form.

A generalized memristor consisting of a single ideal inductor and diode bridge is proposed in [

Model of a generalized fractional-order memristor based on diode bridge cascaded with a single fractional-order inductor.

The relation between the voltage across and the current through diodes

Equivalent unit circuit of the fractional-order inductor.

The electronic circuit shown in Figure

Linear transfer function approximations of fractional-order integrator with order

| |
---|---|

0.99 | |

0.93 | |

0.9 | |

0.8 | |

The equivalent circuit expression of the fractional-order inductor is

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

| | | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|---|

0.99 | 891.25 | 209.451 | 2.172 | 0.023 | 2e(-4) | 1e(-7) | 0.205 | 0.212 | 0.222 | 0.23 | 0.011 |

| |||||||||||

0.93 | 446.7 | 22.6 | 0.31 | 0.004 | 5e(-5) | 2e(-7) | 0.019 | 0.026 | 0.036 | 0.05 | 0.019 |

| |||||||||||

0.9 | 316.2 | 13.7 | 0.22 | 0.003 | 5e(-5) | 3e(-7) | 0.011 | 0.012 | 0.027 | 0.043 | 0.025 |

| |||||||||||

0.8 | 100 | 4.25 | 0.11 | 0.0027 | 6e(-5) | 1e(-6) | 0.003 | 0.007 | 0.017 | 0.042 | 0.064 |

To calculate the approximate transfer functions

The bode diagrams of Oustaloup approximation. (a)

Four diodes 1N4148 and a voltage source

The volt-ampere characteristic of FOM with different orders. (a)

It can be seen that the trajectory is a pinched hysteresis loop. With the same order, the higher the frequency, the smaller the area enclosed by the hysteresis loop. At the same frequency, when the order decreases, the maximum current of the input port increases.

In order to compare the influence of the order on the characteristics of the memristor model, the volt-ampere relationship under different order is obtained as shown in Figure

The volt-ampere characteristic with different

The volt-ampere characteristics of the FOM model obtained in this section conform to the definition of generalized memristor. When

The structure of memristive chaotic circuit, which is a integer-order system, is proposed in [

The FOM-based fractional-order chaotic circuit.

Applying the basic circuit law, the following formulas are obtained:

Make the left side of (

Solution of (

The Jacobian matrix of the equilibrium point is

The fractional-order mathematical model of the generalized FOM-based chaotic circuit has been derived, then numerical simulations can be realized. The simulation of the fractional-order differential equation (

Phase trajectories of fractional-order memristive system at order 0.99; (a) phase trajectory in

Phase trajectory in

Phase trajectory in

Phase trajectory in

Phase trajectory in

The phase trajectory with initial value of (0,0.001,0) is plotted with pink lines, and the trajectory with (0,-0.01,0) is plotted with black lines, and the two colors correspond to two initial conditions in subsequent chapters, respectively. As depicted in Figure

The volt-ampere characteristic curve of the FOM system.

The Poincaré Section of

It should be noted that, in the case of 0.99 order and fixed parameters, the phase trajectory curves of fractional-order circuits under two initial conditions are approximately the same. But with the different order or circuit parameters, the fractional-order circuits will generate more complex dynamic behavior, which will be advanced in the next section.

When the order

Bifurcation diagrams of the variable

The coexisting period limit cycle and the spiral chaotic attractor under different initial conditions in the interval of

System phase trajectories under different orders; (a)

When the order is 0.99 and 0.9, the bifurcation diagram of variable

The bifurcation diagrams under different orders; (a)

Comparing with the integer-order system, in the process from single-period limit cycle to period doubling bifurcation, the state of the system is interrupted once when the order is 1, 7 times when the order is 0.99, and 8 times when the order is 0.9, so the frequency of the system state alternating changes increases with the decrease of the order. One can see that the dynamic behavior of the system from single-period limit cycle to chaos via period doubling bifurcation is similar. But in the whole interval, with the decrease of order, the whole behavior has the tendency of backward delay. The system states under typical parameters are compared as shown in Table

Different system states under different values of

values of | orders | system states |
---|---|---|

| | coexisting upper and lower period limit cycles |

| single limit cycles | |

| single limit cycles | |

| ||

| | single scroll spiral chaotic attractor |

| coexisting upper and lower period limit cycles | |

| single limit cycles | |

| ||

| | multiple period limit cycles |

| single scroll spiral chaotic attractor | |

| single limit cycles | |

| ||

| | double scroll chaotic attractor |

| multiple period limit cycles | |

| coexisting upper and lower period limit cycles |

Figure

Phase trajectories of the system varying with parameter

To compare with the case of integer-order system, Figure

Phase trajectories in

When the order is 0.99 and

Antimonotonic coexisting bubble phenomenon at order 0.99 with different

This phenomenon has been found in some nonlinear systems, such as jerk system [

In this section, circuit simulations are carried out by Pspice. The fractional-order capacitor and inductor in the FOM-based chaotic circuit are realized by equivalent unit circuits. The equivalent circuit of fractional-order inductor has been introduced in forward part, and the equivalent circuit structure of fractional-order capacitor is shown in Figure

Equivalent circuit of fractional-order capacitor.

The equivalent circuit expression of the fractional-order capacitor in complex frequency domain is

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

| | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|

| 0.99 | 1.81e5 | 2.28e4 | 247.51 | 2.71 | 0.03 | 3.3e-4 | 1.6e-7 | 0.04 |

0.88 | 1.21e5 | 8.04e3 | 97.9 | 1.2 | 0.015 | 1.8e-4 | 2.1e-7 | 0.062 | |

| |||||||||

| 0.99 | 7.2e4 | 9.1e4 | 99 | 1.1 | 0.012 | 1.3e-4 | 6.4e-8 | 0.015 |

0.88 | 4.8e4 | 3.2e3 | 39.2 | 0.48 | 0.006 | 7.3e-5 | 8.5e-8 | 0.025 | |

| |||||||||

| 0.99 | 27.6 | 219.3 | 2e4 | 1.8e6 | 1.7e8 | 1.5e10 | 3.1e13 | 1.3e8 |

0.88 | 41.3 | 622 | 5e4 | 4.2e6 | 3.4e8 | 2.7e10 | 2.4e13 | 8e7 |

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

| | | | | | | | |
---|---|---|---|---|---|---|---|---|

| 0.99 | 0.218 | 0.236 | 0.26 | 0.28 | 0.312 | 0.03 | 7.1e3 |

0.88 | 0.073 | 0.088 | 0.11 | 0.135 | 0.164 | 0.038 | 1.1e4 | |

| ||||||||

| 0.99 | 0.09 | 0.09 | 0.1 | 0.11 | 0.125 | 0.012 | 2.8e3 |

0.88 | 0.03 | 0.035 | 0.043 | 0.054 | 0.066 | 0.015 | 4.4e3 | |

| ||||||||

| | | | | | | | |

| ||||||||

| 0.99 | 4.3e-8 | 4.7e-8 | 5.2e-8 | 5.7e-8 | 6.2e-8 | 6e-9 | 0.0014 |

0.88 | 1.4e-8 | 1.8e-8 | 2.2e-8 | 2.7e-8 | 3.3e-8 | 7.6e-9 | 0.002 |

The bode diagrams of Oustaloup approximation of

Figure

Circuit structure diagram Pspice simulation.

The phase trajectory at order 0.99 is shown in Figure

Phase trajectories of variable of fractional-order memristive circuits when

At order 0.88, the circuit presents the limit cycle; the projection on the

Phase trajectories of variable of fractional-order and integer-order memristive circuits; (a) projection on the

In order to compare with the circuit state of integer-order system, Figures

Phase trajectories of variable of fractional-order and integer-order memristive circuits; (a) projection on the

A generalized fractional-order memristor based on fractional-order inductor and diode bridge is proposed and its memristive characteristics are verified. Then a fractional-order chaotic circuit composed of a fractional-order capacitor, inductor, and a negative resistor is proposed. It has an unstable saddle-focus and two unstable node-focuses, which indicates that the system is a double scroll chaotic system with fixed parameters. Then the bifurcation diagram of the system changing with the order is studied. Combining with the system trajectory, under different initial conditions, the system will experience single-period limit cycles, coexisting upper and lower single-period limit cycle, multiple-period limit cycles, and spiral chaos, and finally enters into double scroll chaotic state when the order is less than 1. Then the bifurcation diagrams of the system with one of the parameters at different orders are studied. It presents that the dynamic behavior with same fixed system parameters at different orders is similar to delay process. At the same time, it is found that the fractional-order system has antimonotonic behavior consisting of forward and antiperiodic coexisting upper and lower periodic or chaotic band states. Finally, the chaotic circuit composed of equivalent fractional-order inductor and capacitor is simulated to verify the abundant dynamic behavior of memristive circuit. Compared with other fractional-order memristive chaotic circuits, the chaotic system based on fractional-order memristor has a simpler topology but has abundant dynamic behavior.

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

The authors declare that there are no conflicts 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 Innovation Project (Grant no. 201805037