# Dynamic Behaviors Analysis of a Chaotic Circuit Based on a Novel Fractional-Order Generalized Memristor

Authors: Ningning Yang; Shucan Cheng; Chaojun Wu; Rong Jia; Chongxin Liu
Journal: Complexity
 (2019)

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## Abstract

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.


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## Body

## 1. Introduction

The memristor, which is considered as the fourth basic circuit element, was first proposed theoretically by Professor Leon Chua in 1971[1]. Since the development of the practical memristor in HP Laboratory in 2008 [2], the practical application of memristors has attracted wide attention. At present, the research on the memristor mainly focuses on its physical realization such as memristor equivalent circuits [3–5], the dynamical behaviors of chaotic circuits based on the memristor [6–9], and memristive neural networks [10, 11]. As a basic circuit element, memristor is mostly used in various fields in the form of circuit at present, so the application circuit of memristor is rich and diverse. Because memristor has natural nonlinearity and plasticity, it is easy to construct chaotic oscillation circuit based on memristor by organic combination with other circuit elements.If the volt-ampere characteristics of the circuit ports have three characteristics fingerprints as described in [12], it can be defined as a memristor. Reference [3] presents a generalized memristor diode bridge circuit of cascaded RLC filters consisting of full-wave rectifiers and second-order RLC filters. Reference [4] presents a generalized memristor simulator for first-order parallel RC filters cascaded by a diode bridge. In [5], a first-order generalized memristor simulator based on diode bridge and series RL filter is proposed. In [13], a fractional-order generalized memristor consisting of a diode bridge and a parallel circuit with an equivalent unit circuit and a linear resistance is proposed. If only an inductor cascades a diode bridge, the circuit has simpler topology while it satisfies the memristor characteristics. All the generalized memristor can be simplified as a nonlinear basic element.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 [14, 15], control theory [16], and nonlinear dynamical systems [17–21]. A large number of studies have shown that the introduction of fractional-order parameters as adjustable parameters in the model improves the degree of freedom of the model and can more accurately describe the characteristics of actual systems. Because magnetic flux or charge is mathematically a time integral of voltage or current, it can show itself a memory feature [22]. Fractional calculus is especially suitable for describing the memory and hereditary characteristics of the system [23, 24]. It has essentially the same mathematical principles as the memory characteristics of memory circuit elements. It is feasible to introduce fractional calculus theory into the dynamic behavior analysis and application of memory circuit elements [25, 26]. The relationship between fractional calculus and the behavior of memory system is proposed in [17], and it is pointed out that the memristor can be extended to fractional-order one. Reference [18] describes a classical Chua's oscillator based on fractional-order memristor, in which the memristor is a flux-controlled memristor. A novel fractional-order system consisting of memristor, capacitor, and inductor is studied in [19]. In [27], a memory resistor with noncontact polynomial memory resistance function is used in the simplest chaotic circuit. In addition, a Lorenz system based on fractional-order memristors is proposed in [21], its flux-controlled memristors have different permeance functions, and their dynamic characteristics are analyzed, respectively. However few papers mentioned that the chaotic circuit based on fractional-order memristor has coexisting period and chaotic states under different initial values and has antimonotonic behavior of antiperiodic bifurcation.The rest of this paper mainly includes the following five sections. In Section2, the generalized fractional-order memristor (FOM) based on single fractional-order inductor and diode bridge is proposed and its memristive characteristics are verified. Then a FOM-based fractional-order chaotic circuit is proposed consisting of a fractional-order capacitor, a fractional-order inductor, and a negative resistor. The equilibrium and stability of the circuit are analyzed. In Section 3, the dynamic behavior of the fractional-order system is analyzed. Firstly, the bifurcation diagram of the system with the order parameter is studied, and the changing process of the dynamic behavior is analyzed combined with the system trajectory under different initial conditions. Then the bifurcation diagrams of the system varying with one of the system parameters at different orders are studied. At the same time, it is found that the fractional-order system has antimonotonic behavior of coexistence and antiperiodicity under different initial conditions. In Section 4 the chaotic circuit composed of equivalent fractional-order inductor and capacitor is simulated to verify the dynamic behavior of the fractional-order chaotic circuit. The last part is the conclusion.


## 2. Fractional-Order Chaotic Circuit Based on Generalized Memristor

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 [23].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(1)​aCDtαft=1Γn-αdndtn∫0tfτt-τα-n+1dτwhere n∈N is the first integer which is not less than α. f(t) is a fractional-order derivative of a continuous function of time, which is given in terms of a time integral.Laplace transform is a common tool for describing fractional-order systems. The integer-order Laplace transform can be generalized to the fractional-order form.(2)Ldαxtdtα=snFs-∑k=0m-1skdα-1+kxtdtα-1+kwhere m is an integer value; when the initial value of the system is zero, formula (2) can be simplified as(3)Ldαxtdtα=sαFsWhen t=0, for single-input single-output (SISO) systems, the relationship between the input and output signals x(t) and y(t) is(4)Gs=YsFs=amsβm+am-1sβm-1+···+a0sβ0bmsαn+bm-1sαn-1+···+b0sα0where (am,bm)∈R(∀n,m∈N).


### 2.1. Fractional-Order Memristor Based on Diode Bridge Cascaded with a Single Fractional-Order Inductor

A generalized memristor consisting of a single ideal inductor and diode bridge is proposed in [6]. Lots of researches have shown that the real capacitor and inductor can be extended to fractional-order forms to describe their real electricity characteristics. In this paper, the integer-order inductor L in the generalized memristor is extended to the fractional-order form. Then the model of integer-order memristor can be extended to the fractional-order one as shown in Figure 1.Figure 1
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 diodesDk, named as vk and ik, respectively k=1,2,3,4, is modeled as(5)ik=ISe2ρvk-1where IS is the reverse saturation current of the diode, ρ=1/(2nVT),n is the emission coefficient, VT refers to the thermal voltage, and three diode parameters are 5.84nA, 1.94, and 25mV, respectively. q stands for the order; the fractional-order mathematical model of the circuit shown in Figure 1 is(6)ig=iLm+2IStanh⁡ρvg(7)dqiLmdtq=ln2IScosh⁡ρvg-ln⁡iLm+2ISρLmqiLm refers to the current passing through the fractional-order inductor, vg stands for the input voltage, and ig is the input current. Lmq is set to 10mH. This is a current-controlled FOM model. And the memconductance is(8)GMq=igvg=iLm+2IStanhρvgvgThe fractional-order inductor is realized by equivalent unit circuit shown in Figure 2.Figure 2
Equivalent unit circuit of the fractional-order inductor.The electronic circuit shown in Figure2 is constructed to implement the approximate transfer functions of fractional-order integrator operator depicted in Table 1.Table 1
Linear transfer function approximations of fractional-order integrator with orderq.
q F ( s ) 0.99 s 5 + 9.87 e 4 s 4 + 9.647 e 7 s 3 + 9.43 e 8 s 2 + 9.2 e 7 s + 8.9 e 4 891.3 s 5 + 9.21 e 5 s 4 + 9.43 e 6 s 3 + 9.65 e 5 s 2 + 987.1 s + 0.01  0.93 s 5 + 8.6 e 4 s 4 + 7.32 e 7 s 3 + 6.23 e 8 s 2 + 5.3 e 7 s + 4.47 e 4 446.7 s 5 + 5.3 e 5 s 4 + 6.23 e 6 3 + 7.32 e 5 s 2 + 859.7 s + 0.01  0.9 s 5 + 8.02 e 4 s 4 + 6.37 e 7 s 3 + 5.06 e 8 s 2 + 4.02 e 7 s + 3.2 e 4 316.2 s 5 + 4.02 e 5 s 4 + 5.06 e 6 3 + 6.37 e 5 s 2 + 802.4 s + 0.01  0.8 s 5 + 6.37 e 4 s 4 + 4.02 e 7 s 3 + 2.54 e 8 s 2 + 1.6 e 7 s + 1 e 4 100 s 5 + 1.6 e 5 s 4 + 2.54 e 6 s 3 + 4.02 e 5 s 2 + 637.3 s + 0.01The equivalent circuit expression of the fractional-order inductor is(9)1Ls-q=∑k=1n1/Lks+Rk/Lk+1RinUsing Oustaloup approximation method, the series number of resistors and inductors connected in parallel is n=2N+1, N is the order of the filter. Table 1 gives the approximate transfer functions of fractional-order integrator F(s) when q is selected as 0.99, 0.93, 0.9, and 0.8. The corresponding values of the inductor are calculated in Table 2.Table 2
The inductance parameters of the equivalent chain circuit of the resistor and the fractional-order inductor.
q R i n / Ω R 1 / Ω R 2 / Ω R 3 / Ω R 4 / Ω R 5 / Ω L 1 / H L 2 / H L 3 / H L 4 / H L 5 / H 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 0.064To calculate the approximate transfer functionsFs, the bode diagrams of Oustaloup approximation when q=0.99 and q=0.9 are exhibited in Figure 3.Figure 3
The bode diagrams of Oustaloup approximation. (a)q=0.99; (b) q=0.9.
(a)
q = 0.99 (b)
q = 0.9Four diodes 1N4148 and a voltage sourcevg=Asin(2πft) are selected in the Pspice circuit simulation experiment, where A and f are the amplitude and frequency of the voltage source, respectively, and the equivalent circuit model of the fractional-order inductor is built. When A is 3V and f is 100Hz, 500Hz, and 1 kHz, respectively, the volt-ampere relationship of the input port is shown in Figure 4.Figure 4
The volt-ampere characteristic of FOM with different orders. (a)q=0.99; (b) q=0.93; (c) q=0.9; (d) q=0.8.
(a)
q = 0.99 (b)
q = 0.93 (c)
q = 0.9 (d)
q = 0.8It 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 Figure5.Figure 5
The volt-ampere characteristic with differentq when A=3V,f=500Hz.The volt-ampere characteristics of the FOM model obtained in this section conform to the definition of generalized memristor. Whenf=500Hz, as q decreases, the maximum current of the input port increases and the area of the hysteresis loop increases.


### 2.2. The Fractional-Order Chaotic Circuit Based on FOM

The structure of memristive chaotic circuit, which is a integer-order system, is proposed in [7]. In this paper, the inductor, capacitor, and memristor are extended to fractional order, and then the FOM-based fractional-order chaotic circuit is obtained. The structure of the FOM-based fractional-order chaotic circuit is shown in Figure 6. In addition the case of symmetry order is studied in this paper; that is, the fractional order of the inductor, capacitor, and memristor is maintained as q. When q=1, the system is the integer-order system proposed in [7]. A negative resistor is connected in series by a fractional-order inductor Lq, a fractional-order capacitor Cq and a fractional-order memristor Mq are connected in parallel. vC represents the voltage of Cq; iL and iMq represent the current flowing through Lqand Mq, respectively.Figure 6
The FOM-based fractional-order chaotic circuit.Applying the basic circuit law, the following formulas are obtained:(10)CqdqvCdtq+iMq+iL=0(11)LqdqiLdtq-iLG=vC The mathematical model of FOM is(12)iMq=iLm+2IStanh⁡ρvCdqiLmdtq=ln2IScosh⁡ρvC-lniLm+2ISρLmqThen the fractional-order mathematical model of the chaotic circuit is obtained.(13)dqvCdtq=-iL+iLm+2IStanh⁡ρvCCqdqiLdtq=vC+iL/GLqdqiLmdtq=ln2IScosh⁡ρvC-lniLm+2ISρLmqNormalize the following variables and circuit parameters:(14)v1=ρvC,i1=ρiLG,im=ρiLmG,τ=tGCq,α=CqLmqG2,β=CqLqG2,γ=2ρISGThe normalized fractional-order mathematical model of the chaotic circuit is obtained as follows:(15)dqv1dτq=-i1-im+γtanh⁡v1dqi1dτq=βv1+i1dqimdτq=αln⁡γcosh⁡v1-αln⁡im+γWhen the values of the fractional-order capacitor, inductor, and resistor are set as Cq=5nF, Lq=25mH,  G=0.65mS, and Lmq=10mH, respectively, values of three parameters can be obtained as(16)α=1.1834,β=0.4734,γ=1.8525×10-4


### 2.3. Equilibrium Point and Its Stability

Make the left side of (15) equal to zero,(17)0=-i1-im+γtanh⁡v10=βv1+i10=αln⁡γcosh⁡v1-αln⁡im+γThe equilibrium point can be obtained as(18)O=v~1,-v~1,γcosh⁡v~1-γThe value of v~1 can be obtained by solving transcendental equations(19)γsinh⁡v~1-v~1=0Equation (19) has a zero solution and a pair of symmetric nonzero solutions, then system (15) has a zero equilibrium point O0=(0,0,0) and a pair of nonzero equilibrium points O2,3=(±11.751,∓11.751,11.752). Two nonzero equilibrium points are obtained by parameters in (16). As shown in Figure 7, the blue and orange lines represent the trajectory of x1=γsinh⁡(v1) and x2=v1, respectively. The intersection point is the value of v1.Figure 7
Solution of (19).The Jacobian matrix of the equilibrium point is(20)J=-tanh⁡v~1-1-γsech⁡v~10ββ-αsechv~1γ0αtanh⁡v~1The characteristic equation is(21)Oλ=det1λ-J=λ3+lλ2+mλ+n=0where(22)l=1-αtanh⁡v~1-βm=1-αβtanh⁡v~1+αsech2v~1n=αβtanh2v~1+sech2v~1+sechv~1γThe eigenvalues at three equilibrium points are O0:λ1,2=-7.0869±j12.498,λ3=14.6473, O1,2:λ1=-0.9849,λ2=1.2125,λ3=0.4292. O0 is an unstable saddle-focus which has one positive root and two complex conjugate roots with negative real parts; O1,2 are two unstable node-focuses having two positive roots and one negative root. Therefore, a chaotic orbit can be excited from the unstable saddle-focus O0 and affected by the two unstable node-focuses O1,2, leading to the appearance of double scroll chaotic attractor.


### 2.4. Characteristic Curve of Chaotic Circuit

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 (15) can be carried out by Oustaloup approximation method. Circuit parameters are taken as (16). When the order q is 0.99 and initial values are set to be (0,0.01,0) and (0,-0.01,0), respectively, the chaotic trajectories of system (15) is shown in Figure 8.Figure 8
Phase trajectories of fractional-order memristive system at order 0.99; (a) phase trajectory inv1-i1 plane; (b) phase trajectory in v1-i1 plane; (c) phase trajectory in i1-im plane; (d) phase trajectory in v1-i1-im plane.
(a)
Phase trajectory inv1-i1 plane (b)
Phase trajectory inv1-i1 plane (c)
Phase trajectory ini1-im plane (d)
Phase trajectory inv1-i1-im planeThe 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 Figure8, the chaotic attractor is a double scroll and the fractional-order memristive circuit generates complex attractor topology. Meanwhile, the volt-ampere characteristic curve of the generalized FOM is depicted in Figure 9; the pinched hysteresis loop verifies the validity of the proposed fractional-order generalized memristor. The Poincaré Section of v1-im when i1=0 is depicted in Figure 10, the existence of chaos is verified by regular and closed trajectory.Figure 9
The volt-ampere characteristic curve of the FOM system.Figure 10
The Poincaré Section ofv1-im when i1=0.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.


## 2.1. Fractional-Order Memristor Based on Diode Bridge Cascaded with a Single Fractional-Order Inductor

A generalized memristor consisting of a single ideal inductor and diode bridge is proposed in [6]. Lots of researches have shown that the real capacitor and inductor can be extended to fractional-order forms to describe their real electricity characteristics. In this paper, the integer-order inductor L in the generalized memristor is extended to the fractional-order form. Then the model of integer-order memristor can be extended to the fractional-order one as shown in Figure 1.Figure 1
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 diodesDk, named as vk and ik, respectively k=1,2,3,4, is modeled as(5)ik=ISe2ρvk-1where IS is the reverse saturation current of the diode, ρ=1/(2nVT),n is the emission coefficient, VT refers to the thermal voltage, and three diode parameters are 5.84nA, 1.94, and 25mV, respectively. q stands for the order; the fractional-order mathematical model of the circuit shown in Figure 1 is(6)ig=iLm+2IStanh⁡ρvg(7)dqiLmdtq=ln2IScosh⁡ρvg-ln⁡iLm+2ISρLmqiLm refers to the current passing through the fractional-order inductor, vg stands for the input voltage, and ig is the input current. Lmq is set to 10mH. This is a current-controlled FOM model. And the memconductance is(8)GMq=igvg=iLm+2IStanhρvgvgThe fractional-order inductor is realized by equivalent unit circuit shown in Figure 2.Figure 2
Equivalent unit circuit of the fractional-order inductor.The electronic circuit shown in Figure2 is constructed to implement the approximate transfer functions of fractional-order integrator operator depicted in Table 1.Table 1
Linear transfer function approximations of fractional-order integrator with orderq.
q F ( s ) 0.99 s 5 + 9.87 e 4 s 4 + 9.647 e 7 s 3 + 9.43 e 8 s 2 + 9.2 e 7 s + 8.9 e 4 891.3 s 5 + 9.21 e 5 s 4 + 9.43 e 6 s 3 + 9.65 e 5 s 2 + 987.1 s + 0.01  0.93 s 5 + 8.6 e 4 s 4 + 7.32 e 7 s 3 + 6.23 e 8 s 2 + 5.3 e 7 s + 4.47 e 4 446.7 s 5 + 5.3 e 5 s 4 + 6.23 e 6 3 + 7.32 e 5 s 2 + 859.7 s + 0.01  0.9 s 5 + 8.02 e 4 s 4 + 6.37 e 7 s 3 + 5.06 e 8 s 2 + 4.02 e 7 s + 3.2 e 4 316.2 s 5 + 4.02 e 5 s 4 + 5.06 e 6 3 + 6.37 e 5 s 2 + 802.4 s + 0.01  0.8 s 5 + 6.37 e 4 s 4 + 4.02 e 7 s 3 + 2.54 e 8 s 2 + 1.6 e 7 s + 1 e 4 100 s 5 + 1.6 e 5 s 4 + 2.54 e 6 s 3 + 4.02 e 5 s 2 + 637.3 s + 0.01The equivalent circuit expression of the fractional-order inductor is(9)1Ls-q=∑k=1n1/Lks+Rk/Lk+1RinUsing Oustaloup approximation method, the series number of resistors and inductors connected in parallel is n=2N+1, N is the order of the filter. Table 1 gives the approximate transfer functions of fractional-order integrator F(s) when q is selected as 0.99, 0.93, 0.9, and 0.8. The corresponding values of the inductor are calculated in Table 2.Table 2
The inductance parameters of the equivalent chain circuit of the resistor and the fractional-order inductor.
q R i n / Ω R 1 / Ω R 2 / Ω R 3 / Ω R 4 / Ω R 5 / Ω L 1 / H L 2 / H L 3 / H L 4 / H L 5 / H 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 0.064To calculate the approximate transfer functionsFs, the bode diagrams of Oustaloup approximation when q=0.99 and q=0.9 are exhibited in Figure 3.Figure 3
The bode diagrams of Oustaloup approximation. (a)q=0.99; (b) q=0.9.
(a)
q = 0.99 (b)
q = 0.9Four diodes 1N4148 and a voltage sourcevg=Asin(2πft) are selected in the Pspice circuit simulation experiment, where A and f are the amplitude and frequency of the voltage source, respectively, and the equivalent circuit model of the fractional-order inductor is built. When A is 3V and f is 100Hz, 500Hz, and 1 kHz, respectively, the volt-ampere relationship of the input port is shown in Figure 4.Figure 4
The volt-ampere characteristic of FOM with different orders. (a)q=0.99; (b) q=0.93; (c) q=0.9; (d) q=0.8.
(a)
q = 0.99 (b)
q = 0.93 (c)
q = 0.9 (d)
q = 0.8It 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 Figure5.Figure 5
The volt-ampere characteristic with differentq when A=3V,f=500Hz.The volt-ampere characteristics of the FOM model obtained in this section conform to the definition of generalized memristor. Whenf=500Hz, as q decreases, the maximum current of the input port increases and the area of the hysteresis loop increases.


## 2.2. The Fractional-Order Chaotic Circuit Based on FOM

The structure of memristive chaotic circuit, which is a integer-order system, is proposed in [7]. In this paper, the inductor, capacitor, and memristor are extended to fractional order, and then the FOM-based fractional-order chaotic circuit is obtained. The structure of the FOM-based fractional-order chaotic circuit is shown in Figure 6. In addition the case of symmetry order is studied in this paper; that is, the fractional order of the inductor, capacitor, and memristor is maintained as q. When q=1, the system is the integer-order system proposed in [7]. A negative resistor is connected in series by a fractional-order inductor Lq, a fractional-order capacitor Cq and a fractional-order memristor Mq are connected in parallel. vC represents the voltage of Cq; iL and iMq represent the current flowing through Lqand Mq, respectively.Figure 6
The FOM-based fractional-order chaotic circuit.Applying the basic circuit law, the following formulas are obtained:(10)CqdqvCdtq+iMq+iL=0(11)LqdqiLdtq-iLG=vC The mathematical model of FOM is(12)iMq=iLm+2IStanh⁡ρvCdqiLmdtq=ln2IScosh⁡ρvC-lniLm+2ISρLmqThen the fractional-order mathematical model of the chaotic circuit is obtained.(13)dqvCdtq=-iL+iLm+2IStanh⁡ρvCCqdqiLdtq=vC+iL/GLqdqiLmdtq=ln2IScosh⁡ρvC-lniLm+2ISρLmqNormalize the following variables and circuit parameters:(14)v1=ρvC,i1=ρiLG,im=ρiLmG,τ=tGCq,α=CqLmqG2,β=CqLqG2,γ=2ρISGThe normalized fractional-order mathematical model of the chaotic circuit is obtained as follows:(15)dqv1dτq=-i1-im+γtanh⁡v1dqi1dτq=βv1+i1dqimdτq=αln⁡γcosh⁡v1-αln⁡im+γWhen the values of the fractional-order capacitor, inductor, and resistor are set as Cq=5nF, Lq=25mH,  G=0.65mS, and Lmq=10mH, respectively, values of three parameters can be obtained as(16)α=1.1834,β=0.4734,γ=1.8525×10-4


## 2.3. Equilibrium Point and Its Stability

Make the left side of (15) equal to zero,(17)0=-i1-im+γtanh⁡v10=βv1+i10=αln⁡γcosh⁡v1-αln⁡im+γThe equilibrium point can be obtained as(18)O=v~1,-v~1,γcosh⁡v~1-γThe value of v~1 can be obtained by solving transcendental equations(19)γsinh⁡v~1-v~1=0Equation (19) has a zero solution and a pair of symmetric nonzero solutions, then system (15) has a zero equilibrium point O0=(0,0,0) and a pair of nonzero equilibrium points O2,3=(±11.751,∓11.751,11.752). Two nonzero equilibrium points are obtained by parameters in (16). As shown in Figure 7, the blue and orange lines represent the trajectory of x1=γsinh⁡(v1) and x2=v1, respectively. The intersection point is the value of v1.Figure 7
Solution of (19).The Jacobian matrix of the equilibrium point is(20)J=-tanh⁡v~1-1-γsech⁡v~10ββ-αsechv~1γ0αtanh⁡v~1The characteristic equation is(21)Oλ=det1λ-J=λ3+lλ2+mλ+n=0where(22)l=1-αtanh⁡v~1-βm=1-αβtanh⁡v~1+αsech2v~1n=αβtanh2v~1+sech2v~1+sechv~1γThe eigenvalues at three equilibrium points are O0:λ1,2=-7.0869±j12.498,λ3=14.6473, O1,2:λ1=-0.9849,λ2=1.2125,λ3=0.4292. O0 is an unstable saddle-focus which has one positive root and two complex conjugate roots with negative real parts; O1,2 are two unstable node-focuses having two positive roots and one negative root. Therefore, a chaotic orbit can be excited from the unstable saddle-focus O0 and affected by the two unstable node-focuses O1,2, leading to the appearance of double scroll chaotic attractor.


## 2.4. Characteristic Curve of Chaotic Circuit

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 (15) can be carried out by Oustaloup approximation method. Circuit parameters are taken as (16). When the order q is 0.99 and initial values are set to be (0,0.01,0) and (0,-0.01,0), respectively, the chaotic trajectories of system (15) is shown in Figure 8.Figure 8
Phase trajectories of fractional-order memristive system at order 0.99; (a) phase trajectory inv1-i1 plane; (b) phase trajectory in v1-i1 plane; (c) phase trajectory in i1-im plane; (d) phase trajectory in v1-i1-im plane.
(a)
Phase trajectory inv1-i1 plane (b)
Phase trajectory inv1-i1 plane (c)
Phase trajectory ini1-im plane (d)
Phase trajectory inv1-i1-im planeThe 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 Figure8, the chaotic attractor is a double scroll and the fractional-order memristive circuit generates complex attractor topology. Meanwhile, the volt-ampere characteristic curve of the generalized FOM is depicted in Figure 9; the pinched hysteresis loop verifies the validity of the proposed fractional-order generalized memristor. The Poincaré Section of v1-im when i1=0 is depicted in Figure 10, the existence of chaos is verified by regular and closed trajectory.Figure 9
The volt-ampere characteristic curve of the FOM system.Figure 10
The Poincaré Section ofv1-im when i1=0.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.


## 3. Dynamic Behavior Analysis of the Fractional-Order Memristor Chaotic System

### 3.1. The Influence of the Order on the Dynamic Behavior of the System

When the orderq varies between regions [0.8,1], the bifurcation diagram for the variable v1 versus q is shown in Figure 11.Figure 11
Bifurcation diagrams of the variablev1 versus q.The coexisting period limit cycle and the spiral chaotic attractor under different initial conditions in the interval of[0.895,0.947] are a phenomenon worthy of further investigation. For the same initial conditions, the maxima value of the up and down jump variables results in a discontinuity in dynamic behavior. And the dynamic behavior under two initial conditions just happens to form a coexisting period phenomenon. After entering the double scroll chaos, the trajectories are interlaced into the same closed region. The phase trajectory of the system from q=0.88 to q=0.962 is shown in Figure 12. It reveals that the system has experienced a single-period limit cycle, coexisting upper and lower single-period limit cycles, double-period limit cycles, and spiral chaotic attractors, and finally enters the double scroll chaotic attractor. Figure 12(f) shows a multiperiod state occurring in a narrow periodic window at order 0.962. Further, in Figure 12(a), the pink trajectory is covered by the black trajectory, so the pink trajectory is not seen, and the pink and black trajectories are interlaced in Figure 12(f), so that the coexisting phenomenon is not seen. The chaotic phase diagram further confirms that the fractional-order chaotic circuit based on the FOM can generate rich dynamic behavior.Figure 12
System phase trajectories under different orders; (a)q=0.88; (b) q=0.9; (c) q=0.93; (d) q=0.942; (e) q=0.955; (f) q=0.962.
(a)
q = 0.88 (b)
q = 0.9 (c)
q = 0.93 (d)
q = 0.942 (e)
q = 0.955 (f)
q = 0.962


### 3.2. The Influence of System Parameters on System Dynamics Behaviors

When the order is 0.99 and 0.9, the bifurcation diagram of variablev1 varying with system parameter β is shown in Figure 13. It can be seen that the system moves from the single-period limit cycle to the coexisting upper and lower period limit cycle, then from the coexisting upper and lower four-period limit cycle to the coexisting upper and lower spiral chaotic attractor through period doubling bifurcation, and finally enters the complete chaos. But on the whole, the coexisting bifurcation mode of the system when the order is 0.99 lags behind the one when the order is 0.9.Figure 13
The bifurcation diagrams under different orders; (a)q=0.99; (b)q=0.9.
(a)
q = 0.99 (b)
q = 0.9Comparing 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 Table3.Table 3
Different system states under different values ofβ and q.
values ofβ orders system states β = 0.21 q = 1 coexisting upper and lower period limit cycles q = 0.99 single limit cycles q = 0.9 single limit cycles  β = 0.31 q = 1 single scroll spiral chaotic attractor q = 0.99 coexisting upper and lower period limit cycles q = 0.9 single limit cycles  β = 0.36 q = 1 multiple period limit cycles q = 0.99 single scroll spiral chaotic attractor q = 0.9 single limit cycles  β = 0.56 q = 1 double scroll chaotic attractor q = 0.99 multiple period limit cycles q = 0.9 coexisting upper and lower period limit cyclesFigure14 reveals that the trajectory of system (15) undergoes coexisting upper and lower single-period limit cycle, coexisting upper and lower double-period limit cycles, coexisting upper and lower four-period limit cycles, coexisting spiral chaotic attractors, multiperiod limit cycles in narrow periodic windows, and complete chaos.Figure 14
Phase trajectories of the system varying with parameterβ at order 0.9; (a) β=0.5; (b) β=0.6; (c) β=0.605; (d) β=0.62; (e) β=0.7; (f) β=0.8.
(a)
β = 0.5 (b)
β = 0.6 (c)
β = 0.605 (d)
β = 0.62 (e)
β = 0.7 (f)
β = 0.8To compare with the case of integer-order system, Figure15 shows phase diagrams in v1-i1-im space for different β at q=1 and q=0.9, in which the systems are double scroll chaotic states in integer-order system (Figures 15(a) and 15(b)). And the systems are double-period limit cycles at β=0.6 (Figure 15(c)) and spiral attractor at β=0.62 (Figure 15(d)) in fractional-order system. Figures 14 and 15 are consistent with the bifurcation diagram of Figure 13(b).Figure 15
Phase trajectories inv1-i1-im space of integer-order and fractional-order memristive system; (a) q=1, β=0.6; (b) q=1, β=0.62; (c) q=0.9, β=0.6; (d) q=0.9, β=0.62.
(a)
q = 1, β=0.6 (b)
q = 1, β=0.62 (c)
q = 0.9, β=0.6 (d)
q = 0.9, β=0.62


### 3.3. Phenomenon of Antimonotonic Coexisting Bubbles

When the order is 0.99 andα takes different values, the bifurcation diagram of the system variable v1 varying with the parameter β is shown in Figure 16, in which the variation interval of β is [0.15,0.6]. The phenomenon of antimonotone coexisting bubbles is revealed.Figure 16
Antimonotonic coexisting bubble phenomenon at order 0.99 with differentα; (a) α=0.7; (b) α=0.76; (c) α=0.779; (d) α=0.8; (e) α=0.87.
(a)
α = 0.7 (b)
α = 0.76 (c)
α = 0.779 (d)
α = 0.8 (e)
α = 0.87This phenomenon has been found in some nonlinear systems, such as jerk system [28–30] and Chua's circuit [31]. There is an antimonotonic coexisting upper and lower periodic limit cycle bubble at α=0.7, and there is a coexisting upper and lower double-period bubble in the interval [0.3,0.5] at α=0.76, and there are coexisting upper and lower double-period and four-period bubbles at α=0.779, then the system shows a coexisting upper and lower chaotic band and finally returns to the single-period limit cycle through an antidoubling period bifurcation at α=0.8, and then there are no antimonotonic coexistence bubbles at α=0.87.


## 3.1. The Influence of the Order on the Dynamic Behavior of the System

When the orderq varies between regions [0.8,1], the bifurcation diagram for the variable v1 versus q is shown in Figure 11.Figure 11
Bifurcation diagrams of the variablev1 versus q.The coexisting period limit cycle and the spiral chaotic attractor under different initial conditions in the interval of[0.895,0.947] are a phenomenon worthy of further investigation. For the same initial conditions, the maxima value of the up and down jump variables results in a discontinuity in dynamic behavior. And the dynamic behavior under two initial conditions just happens to form a coexisting period phenomenon. After entering the double scroll chaos, the trajectories are interlaced into the same closed region. The phase trajectory of the system from q=0.88 to q=0.962 is shown in Figure 12. It reveals that the system has experienced a single-period limit cycle, coexisting upper and lower single-period limit cycles, double-period limit cycles, and spiral chaotic attractors, and finally enters the double scroll chaotic attractor. Figure 12(f) shows a multiperiod state occurring in a narrow periodic window at order 0.962. Further, in Figure 12(a), the pink trajectory is covered by the black trajectory, so the pink trajectory is not seen, and the pink and black trajectories are interlaced in Figure 12(f), so that the coexisting phenomenon is not seen. The chaotic phase diagram further confirms that the fractional-order chaotic circuit based on the FOM can generate rich dynamic behavior.Figure 12
System phase trajectories under different orders; (a)q=0.88; (b) q=0.9; (c) q=0.93; (d) q=0.942; (e) q=0.955; (f) q=0.962.
(a)
q = 0.88 (b)
q = 0.9 (c)
q = 0.93 (d)
q = 0.942 (e)
q = 0.955 (f)
q = 0.962


## 3.2. The Influence of System Parameters on System Dynamics Behaviors

When the order is 0.99 and 0.9, the bifurcation diagram of variablev1 varying with system parameter β is shown in Figure 13. It can be seen that the system moves from the single-period limit cycle to the coexisting upper and lower period limit cycle, then from the coexisting upper and lower four-period limit cycle to the coexisting upper and lower spiral chaotic attractor through period doubling bifurcation, and finally enters the complete chaos. But on the whole, the coexisting bifurcation mode of the system when the order is 0.99 lags behind the one when the order is 0.9.Figure 13
The bifurcation diagrams under different orders; (a)q=0.99; (b)q=0.9.
(a)
q = 0.99 (b)
q = 0.9Comparing 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 Table3.Table 3
Different system states under different values ofβ and q.
values ofβ orders system states β = 0.21 q = 1 coexisting upper and lower period limit cycles q = 0.99 single limit cycles q = 0.9 single limit cycles  β = 0.31 q = 1 single scroll spiral chaotic attractor q = 0.99 coexisting upper and lower period limit cycles q = 0.9 single limit cycles  β = 0.36 q = 1 multiple period limit cycles q = 0.99 single scroll spiral chaotic attractor q = 0.9 single limit cycles  β = 0.56 q = 1 double scroll chaotic attractor q = 0.99 multiple period limit cycles q = 0.9 coexisting upper and lower period limit cyclesFigure14 reveals that the trajectory of system (15) undergoes coexisting upper and lower single-period limit cycle, coexisting upper and lower double-period limit cycles, coexisting upper and lower four-period limit cycles, coexisting spiral chaotic attractors, multiperiod limit cycles in narrow periodic windows, and complete chaos.Figure 14
Phase trajectories of the system varying with parameterβ at order 0.9; (a) β=0.5; (b) β=0.6; (c) β=0.605; (d) β=0.62; (e) β=0.7; (f) β=0.8.
(a)
β = 0.5 (b)
β = 0.6 (c)
β = 0.605 (d)
β = 0.62 (e)
β = 0.7 (f)
β = 0.8To compare with the case of integer-order system, Figure15 shows phase diagrams in v1-i1-im space for different β at q=1 and q=0.9, in which the systems are double scroll chaotic states in integer-order system (Figures 15(a) and 15(b)). And the systems are double-period limit cycles at β=0.6 (Figure 15(c)) and spiral attractor at β=0.62 (Figure 15(d)) in fractional-order system. Figures 14 and 15 are consistent with the bifurcation diagram of Figure 13(b).Figure 15
Phase trajectories inv1-i1-im space of integer-order and fractional-order memristive system; (a) q=1, β=0.6; (b) q=1, β=0.62; (c) q=0.9, β=0.6; (d) q=0.9, β=0.62.
(a)
q = 1, β=0.6 (b)
q = 1, β=0.62 (c)
q = 0.9, β=0.6 (d)
q = 0.9, β=0.62


## 3.3. Phenomenon of Antimonotonic Coexisting Bubbles

When the order is 0.99 andα takes different values, the bifurcation diagram of the system variable v1 varying with the parameter β is shown in Figure 16, in which the variation interval of β is [0.15,0.6]. The phenomenon of antimonotone coexisting bubbles is revealed.Figure 16
Antimonotonic coexisting bubble phenomenon at order 0.99 with differentα; (a) α=0.7; (b) α=0.76; (c) α=0.779; (d) α=0.8; (e) α=0.87.
(a)
α = 0.7 (b)
α = 0.76 (c)
α = 0.779 (d)
α = 0.8 (e)
α = 0.87This phenomenon has been found in some nonlinear systems, such as jerk system [28–30] and Chua's circuit [31]. There is an antimonotonic coexisting upper and lower periodic limit cycle bubble at α=0.7, and there is a coexisting upper and lower double-period bubble in the interval [0.3,0.5] at α=0.76, and there are coexisting upper and lower double-period and four-period bubbles at α=0.779, then the system shows a coexisting upper and lower chaotic band and finally returns to the single-period limit cycle through an antidoubling period bifurcation at α=0.8, and then there are no antimonotonic coexistence bubbles at α=0.87.


## 4. Circuit Simulation Experiment

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 Figure17.Figure 17
Equivalent circuit of fractional-order capacitor.The equivalent circuit expression of the fractional-order capacitor in complex frequency domain is(23)Fs=1Cs-q=Rin+∑k=1n1/Cks+1/RkCkOrder q is set as 0.99 and 0.88; Lq, Lmq, and Cq in Figure 6 are selected as 25mH, 10mH, and 5nF. By Oustaloup approximation technique, the corresponding values of inductance and resistance are calculated in Tables 4 and 5. The bode diagrams of Oustaloup approximation of Cq=5nF and Lq=25mH are shown in Figure 18.Table 4
The resistance of the equivalent chain circuit of fractional-order inductor and capacitor.
q R i n ( Ω ) R 1 ( Ω ) R 2 ( Ω ) R 3 ( Ω ) R 4 ( Ω ) R 5 ( Ω ) R 6 ( Ω ) R 7 ( Ω ) L q  (25mH) 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  L m q  (10mH) 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  C q  (5nF) 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 8e7Table 5
The parameters of the equivalent chain circuit of fractional-order inductor and capacitor.
q L 1 ( H ) L 2 ( H ) L 3 ( H ) L 4 ( H ) L 5 ( H ) L 6 ( H ) L 7 ( H ) L q  (25mH) 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  L m q  (10mH) 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  q C 1 ( F ) C 2 ( F ) C 3 ( F ) C 4 ( F ) C 5 ( F ) C 6 ( F ) C 7 ( F )  C q  (5nF) 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 0.002Figure 18
The bode diagrams of Oustaloup approximation ofCq. (a) Lq=25mH, q=0.99; (b) Lq=25mH, q=0.88; (c) Cq=5nF, q=0.99; (d) Cq=5nF, q=0.88.
(a)
L q = 25 m H ,   q = 0.99 (b)
L q = 25 m H ,   q = 0.88 (c)
C q = 5 n F ,   q = 0.99 (d)
C q = 5 n F ,   q = 0.88Figure19 is the circuit structure diagram of simulation in Pspice. The calculated resistance, inductance, and capacitance are used to form equivalent fractional-order inductor and capacitor, respectively. In addition, four diodes 1N4148 and a 1538.5Ω negative resistance which is used to replace conductor of 0.65mS are selected to form fractional-order chaotic circuit in Figure 6.Figure 19
Circuit structure diagram Pspice simulation.The phase trajectory at order 0.99 is shown in Figure20, where the circuit operates in a double scroll chaotic state.Figure 20
Phase trajectories of variable of fractional-order memristive circuits whenq=0.99; (a) projection on the vC-iL plane; (b) projection on the vC-iLm plane; (c) projection on the iL-iLm plane; (d) projection on the vC-iMq plane.
(a)
v C - i L (b)
v C - i L m (c)
i L - i L m (d)
v C - i M qAt order 0.88, the circuit presents the limit cycle; the projection on thevC-iL and vC-iLm plane is shown in Figures 21(a) and 21(b), respectively.Figure 21
Phase trajectories of variable of fractional-order and integer-order memristive circuits; (a) projection on thevC-iL plane when q=0.88; (b) projection on the vC-iLm plane when q=0.88.
(a)
v C - i L (b)
v C - i L mIn order to compare with the circuit state of integer-order system, Figures22(a) and 22(b) are projection on the vC-iL and vC-iLm plane under the same default initial conditions when q=1. The circuit simulation is in good agreement with the theoretical and numerical analysis. It is proved that the chaotic circuit based on the generalized fractional-order memristor has abundant dynamic behavior.Figure 22
Phase trajectories of variable of fractional-order and integer-order memristive circuits; (a) projection on thevC-iL plane when q=1; (b) projection on the vC-iLm plane when q=1.
(a)
v C - i L (b)
v C - i L m


## 5. Conclusion

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.



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*Source: 6083853-2019-05-21.xml*