Cite this Article
Wang Guang-Yi, Jin Pei-Pei, Wang Xiao-Wei, Shen Yi-Ran, Yuan Fang, Wang Xiao-Yuan. A flux-controlled model of meminductor and its application in chaotic oscillator. Chinese Physics B, 2016, 25(9): 090502  
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A flux-controlled model of meminductor and its application in chaotic oscillator
Wang Guang-Yi1, Jin Pei-Pei1, Wang Xiao-Wei2, †, , Shen Yi-Ran1, Yuan Fang1, Wang Xiao-Yuan1
Key Laboratory of RF Circuits and Systems (Ministry of Education), Institute of Modern Circuits and Intelligent Information, Hangzhou Dianzi University, Hangzhou 310018, China
Department of Automation, Shanghai University, Shanghai 200072, China

 

† Corresponding author. E-mail: laura423_wang@163.com

Project supported by the National Natural Science Foundation of China (Grant Nos. 61271064, 61401134, and 60971046), the Natural Science Foundation of Zhejiang Province, China (Grant Nos. LZ12F01001 and LQ14F010008), and the Program for Zhejiang Leading Team of S&T Innovation, China (Grant No. 2010R50010).

Abstract
Abstract

A meminductor is a new type of memory device developed from the memristor. We present a mathematical model of a flux-controlled meminductor and its equivalent circuit model for exploring the properties of the meminductor in a nonlinear circuit. We explore the response characteristics of the meminductor under the exciting signals of sinusoidal, square, and triangular waves by using theoretical analysis and experimental tests, and design a meminductor-based oscillator based on the model. Theoretical analysis and experiments show that the meminductor-based oscillator possesses complex bifurcation behaviors and can generate periodic and chaotic oscillations. A special phenomenon called the co-existent oscillation that can generate multiple oscillations (such as chaotic, periodic oscillations as well as stable equilibrium) with the same parameters and different initial conditions occurs. We also design an analog circuit to realize the meminductor-based oscillator, and the circuit experiment results are in accordance with the theory analysis.

PACS: 05.45.–a;05.45.Jn;05.45.Pq
Keyword:meminductor;oscillator;chaos;co-existent attractor
1. Introduction

As a novel memory device, the memristor was proposed by Chua as early as 1971.[1] Since Strukov et al. realized a physical memristor by nanotechnology,[2] the research of the memristor and its application circuits has gained rapid progress world wide. The meminductor and memcapacitor are members of a large family of new circuit elements postulated by Chua in his guest lecture at the 1978 European Conference on Circuit Theory and Design (ECCTD).[3] The meminductor and memcapacitor were formally defined by Chua in 2003.[4] In particular, the memristor can be identified as a higher-order element (−1, −1), the meminductor can be identified as a higher-order element (−2, −1), and the memcapacitor can be identified as a higher-order element (−1, −2). More recently, the meminductor and memcapacitor were presented, along with their standard symbols, at the keynote lecture of the First Symposium on Memristors, held in Berkeley, California in 2008.[5] A tutorial on the memcapacitor and meminductor was presented in the last section of the IEEE Expert Now SHORT Course.[6] Readers are encouraged to read the tutorial in order to have a deeper understanding of the present paper.

The memristor, memcapacitor, and meminductor are all passive memory devices which can store information without a power supply. Because of their unique memory and dynamical storage ability, they can be used in the fields such as non-volatile storage and simulations of learning, adaptive, and spontaneous behaviors.[7,8] In the meantime, the applications of these memory devices in nonlinear circuits has gained a great deal of attention.

As the representative of memory devices, the memristor appeared earliest in nonlinear circuits. At present, the models of the memristor mostly adopt a piecewise linear, quadratic, or cubic smooth function[9–11] as the charge–flux relationship. All the above models are ideal, and the HP TiO2 physical model is more practical to the needs of applications. Compared with the memristor, the memcapacitor and meminductor have been much less studied because they have not yet been realized physically and their mathematical models are not as perfect as that of the memristor.[12] So we do not know much about their properties in nonlinear circuits.

Although the physical meminductor has not appeared, its potential value has attracted many researchers. In 2011, the Pershin group[13] reported the accomplishment of modeling the memristor, meminductor, and memcapacitor. Then the simulation of the meminductor was realized by using controlled sources and programed modules.[14] Further, a circuit-easily realized equivalent circuit model was proposed by using the transfer relationship between the memristor and meminductor.[15] In Ref. [16], a mathematical model of the meminductor was proposed by using the fraction calculus. In Ref. [17], an approximate equivalent circuit of a meminductor imitated by a memristor was presented. In Ref. [18], an easily realized cubic-smooth active flux-controlled memristor instead of the digital memristor circuit in Ref. [17] was designed. An analysis of the meminductor properties was carried out and an approximate meminductor equivalent circuit simulation and experimental verification were done by using the equivalent circuit of a memristor. Furthermore, in Ref. [19], an efficient method was proposed to build a mutator-based meminductor (ML) model whose inductance can be varied by an external current source.

From the above, we can see that these meminductor models are all ideal. In fact, the memristive behaviors are more readily observable and significant at the nanoscale,[5,20,21] and the properties of the memristor, meminductor, and memcapacitor are common at the nanoscale.[5] Hence, in this paper, we propose an HP memristor-like meminductor model, which conforms more to the future achievable physical one. Based on the proposed flux-controlled model of a meminductor, an equivalent circuit for imitating the behaviors of the meminductor model is designed and realized. The responses for the excitations of waves of sine, pulse, and triangle are verified respectively via the equivalent circuit experiments of the meminductor model. Based on the model, a chaotic oscillator circuit is designed and physically realized. The dynamical behaviors of the oscillator circuit such as periodic oscillation, chaotic oscillation, and their bifurcations are analyzed by theoretical analysis, simulation and circuit experiments, and the results of the experiments are in accordance with the theory analysis and simulations.

2. Models of flux-controlled meminductor
2.1. Mathematical model

It has been found that some microsystems exhibit meminductive properties.[5,20,21] Hence, we can deduce a mathematical model of a meminductor from the nanoscale HP memristor model, and it can be called the HP memristor-like model.

According to Ref. [2], the HP TiO2 memristor has the following model:

where u(t), i(t), and q(t) denote the voltage, current, and charge across the memristor; Ron and Roff stand for the resistances of the high doping domain and the low doping domain, respectively; D and w represent the thicknesses of the whole domain between two boards of platinum electodes and the doping TiO2 layer, respectively; and μv denotes the dopant mobility. According to Eq. (1), the resistance of the memristor can be calculated as

Let α = Roff and β = (Ron − Roff)uvRon/D2, we obtain the simplified model as

Memristance and memductance follow the same law except they are reciprocal. So we can write down the mathematical expression of memductance according to Eq. (4)

where φ(t) is the flux across the memristor.

A meminductor, by contrast to a memresistor, has the definition that it is a two-port passive device whose properties depend on the state and its history. The current through the meminductor relate to its flux, typically showing a pinched hysteretic loop between the flux and the current.

Ventra et al. defined the current-controlled and the flux-controlled meminductors,[5] which are described as follows:

where x is an internal state variable, φ(t) and i(t) are the flux and the current across the meminductor, and LM and are the meminductance and the inverse meminductance, respectively. From the above, we can conclude that both LM and are time-varying functions of current and flux. Analogous to Eqs. (1)–(5), we obtain the simplified model of the meminductor as

We usually adopt the form of inverse meminductance in practice

where is the integral variable of the flux. By integrating two sides of Eq. (8), we obtain the mathematical model of the meminductor

which defines the relationship between the charge q(t) and the integral variable ρ(t).

Let α = 0.03, β = 0.36, and φ(t) = Φm sin(2πFt), according to Eq. (8), the current through the meminductor can be described as

where and .

The time domain waveforms of the current and the flux are shown in Fig. 1. Let Φm = 1 Wb, Φm = 0.8 Wb, and Φm = 0.5 Wb, based on Eq. (8), we obtain the typical pinched hysteretic loops in the two constitutive variables, flux vs. current, as shown in Fig. 1(b); and let Φm = 1 Wb, the phase portrait of ρ and q is pictured in Fig. 1(c) based on Eq. (10).

Fig. 1. (a) Waveforms of current and flux for the flux-controlled meminductor; (b) pinched hysteresis loop of flux-current; (c) ρ–q characteristic.
2.2. Equivalent circuit model

An equivalent circuit is important for the meminductor research since a physical meminductor has not been realized. The equivalent circuit model based on Eq. (8) is shown in Fig. 2, which is composed of five operational amplifiers and one multiplier. The operational amplifiers are all LF347 and the multiplier is AD633. Voltage u is the input of the equivalent circuit. Nodes i and φ are the test points of the current and the flux, respectively. From the output of the inverse integrator constructed by U1, C1, R1, and R2, we can obtain the flux signal . Then the flux −φ(t) passes through the inverter consisting of U2, R3, and R5, and φ(t) is obtained from its output as the input of the adder consisting of U4 and R7–R9. In the mean time, the output, −φ(t), of U1 passes through the inverse integrator constructed by U3, C2, R4, and R6, and then the integral signal is obtained from the output. The signal ρ(t) passes through the multiplier and the product term ρ(t)φ(t) is obtained. Then the product term is sent to the input of the adder. Finally, at the output of U4, we obtain the current

which realize the relationship between current i(t) and flux φ(t) for the meminductor. Comparing with Eq. (10), we have α = R11R9/R10R7 and β = R11R9/R10R8.

Fig. 2. Equivalent circuit of the flux-controlled memindutor with φ–i property.

We have built an equivalent circuit to emulate the features of the meminductor. Now the feasibility of the circuit is proved with hardwire experiments. Figure 3 shows the oscilloscope displays of i (green)–φ (yellow) waveforms and the corresponding pinched hysteresis loops from the hardwire experiments, in which the input signal is sine voltage Um sin2π ft with different frequencies. Figure 4 shows the experimental waveforms of current i (green) and voltage u (yellow) as well as the corresponding pinched hysteresis loops from the hardwire experiments with the sine input Um sin2π ft.

Fig. 3. Experimental current (green) and flux (yellow) waveforms and the corresponding pinched hysteresis loops for the equivalent meminductor circuit with a sinusoidal input at the frequency of: (a), (b) 10 Hz; (c), (d) 25 Hz; (e), (f) 70 Hz.
Fig. 4. Experimental current (green) and voltage (yellow) waveforms and the corresponding phase diagrams for the equivalent meminductor circuit with a sinusoidal input at the frequency of: (a), (b) 10 Hz; (c), (d) 25 Hz; (e), (f) 70 Hz.

From Figs. 3 and 4, we can see that the pinched hysteresis loops defining the meminductor must pass through the origin of the flux vs. current plane under all bipolar periodic excitations in either flux or current, and gradually become a straight line with the increase of the sine frequency, as shown in Figs. 3(b), 3(d), and 3(f). However, the phase portrait of the current vs. voltage for the same meminductor may not pass through the origin, and can be a circle instead of a pinched hysteresis loop, as shown in Figs. 4(b), 4(d), and 4(f). The frequency dependence of the pinched hysteresis loop is a unique feature of the meminductor.

When we use a square-wave voltage to drive the equivalent meminductor circuit, the experimental waveforms of current i (green) and flux φ (yellow) and the corresponding pinched hysteresis loops from the hardwire experiment at different square-wave frequencies are shown in Fig. 5, and the corresponding experimental waveforms and phase portraits for current i and voltage u are shown in Fig. 6.

Fig. 5. Experimental current (green) and flux (yellow) waveforms and the corresponding pinched hysteresis loops for the equivalent meminductor circuit with a square-wavel input at the frequency of: (a), (b) 10 Hz; (c), (d) 25 Hz; (e), (f) 70 Hz.
Fig. 6. Experimental current (green) and voltage (yellow) waveforms and the corresponding phase diagrams for the equivalent meminductor circuit with a square-wavel input at the frequency of: (a), (b) 10 Hz; (c), (d) 25 Hz; (e), (f) 70 Hz.

Figures 7 and 8 show the experimental results with a triangle input voltage at different frequency for the current i (green)–flux φ (yellow) and the current i (green)–voltage u (yellow), respectively.

Fig. 7. Experimental current (green) and flux (yellow) waveforms and the corresponding pinched hysteresis loops for the equivalent meminductor circuit with a triangle input voltage at the frequency of: (a), (b) 10 Hz; (c), (d) 25 Hz; (e), (f) 70 Hz.
Fig. 8. Experimental current (green) and voltage (yellow) waveforms and the corresponding phase diagrams for the equivalent meminductor circuit with a triangle input voltage at the frequency of: (a), (b) 10 Hz; (c), (d) 25 Hz; (e), (f) 70 Hz.

Note again that although the pinched hysteresis loops must pass through the origin of the flux vs. current plane under bipolar periodic excitation in either flux or current, the phase portrait of the current vs. voltage can be a circle not passing through the origin, as shown in Figs. 5–8.

2.3. Equivalent circuit model of the ρ–q property for the meminductor

The equivalent circuit model of the ρ–q property for the meminductor based on Eq. (10) is shown in Fig. 9, which includes three operational amplifiers and one multiplier.

Fig. 9. Equivalent circuit of the flux-controlled meminductor with the ρ–q property.

The input signal ρ, which equals the time integral of the flux, becomes ρ2 after passing the multiplier A2, and then it becomes −ρ2 at the output of U6. In another terminal, ρ passes through the inverter consisting of U5, R10, and R11, and then we obtain −ρ from its output. Finally, at the output of the adder composed of U7, R15, and R16, we obtain

which realizes the ρ–q relationship described by Eq. (10). The phase diagram of flux ρ vs. charge q is described in Fig. 10, which is obtained from the simulation using Multisim.

Fig. 10. Phase diagram of ρ–q.
3. Meminductor-based oscillator
3.1. Oscillator circuit

Reference [22] reported the simplest chaotic circuit, which uses only a linear passive inductor, a linear passive capacitor, and a nonlinear active memristor, and shows several attractors. Here, we present an oscillator circuit in Fig. 11 based on the proposed meminductor model in this paper. The oscillator circuit is composed of a resistor, a negative resistor labeled with the negative conductance −G, two capacitors, and a nonlinear element, namely, the flux-controlled meminductor. The negative resistor is used to supply the circuit with energy for maintaining the oscillation.

Fig. 11. Meminductor-based oscillator circuit.

We take capacitor voltages u1, u2 and flux φ of the meminductor as the state variables. By applying Kirchhoff’s laws to this circuit, the governing equations are obtained as

By inserting the model iM = [α + βρ(t)]φ(t) and adding the equation dρ/dt = φ into Eq. (11) respectively, and setting a = α/C1, b = β/C1, c = 1/(C1R), d = G/C2, e = α/C2, and f = β/C2, equation (11) can be given as

With suitable parameters, this circuit can exhibit periodic and chaotic oscillations. For example, with a = 1.0, b = 6.0, c = 4.6, d = 0.4, e = 4.0, and f = 1.0, the oscillator generates a periodic signal; while if d = 0.7 (the other parameters stay the same), the oscillator is in a chaotic state. The generated periodic and chaotic waveforms of voltage u2 are shown in Fig. 12, and the chaotic phase portraits, namely, the chaotic attractors, are described in Fig. 13.

Fig. 12. (a) Periodic voltage u2 generated from the oscillator; (b) chaotic voltage u2.
Fig. 13. Chaotic phase portraits for: (a) u1–u2; (b) u1–φ; (c) u1–ρ; (d) u2–φ; (e) u2–ρ; (f) φ–ρ.
3.2. Co-existent oscillation and co-existent chaotic attractor

The co-existent oscillation is the oscillation that can generate different types of signals with different initial values of the state variables under the same circuit parameters. The so-called different types of signals are periodic, quasi-periodic, chaotic signals as well as equilibrium points. The attractor generated from the co-existent oscillation is called a co-existent attractor.[23,24]

There are two types of co-existent oscillations. One is the homogeneous oscillations which have the same oscillation types but with different trajectories in their phase space, such as two chaotic oscillations started from two different sets of initial values with the same parameters. Another is the inhomogeneous oscillations which have different oscillation types with different initial values, such as periodic and chaotic oscillations started from two different sets of initial values with the same parameters. Our analysis indicates that the dynamical states of Eq. (12) depend on the setting of the initial conditions, that is, it has multi-stable modes under different initial conditions. So the system can turn from a chaotic state to a periodic state or even a stable equilibrium point when only its initial conditions are changed. When the parameters are set to be a = 1.0, b = 6.0, c = 4.5, d = 0.77, e = 4.0, and f = 1.0, the Lyapunov exponent spectrum with changing u1(0) in the initial condition (u1(0),0.01,0.01,0.01) and the corresponding bifurcation diagram are shown in Fig. 14.

Fig. 14. (a) Lyapunov exponent spectrum with initial value u1 (0); (b) the corresponding bifurcation diagram.

From Fig. 14, we can see that as the initial value u1(0) is varied, the system shows different dynamical behaviors, that is, co-existent attractors appear. In particular, when u1(0) is varied in the range [0, 1.8], the system is in a chaotic state with a positive Lyapunov exponent; with u1(0) in the range [1.8, 9.2], the system is in a periodic state with a maximal zero Lyapunov exponent; when u1(0) is varied in the range [9.2, 10], all the Lyapunov exponents are negative, implying that it is in a stable state. Note that the system undergoes an inverse period-doubling bifurcation in the periodic region of u1(0).

Figure 15 describes the evolution process of several typical attractors, and Table 1 shows the initial values used. Similarly, when u2(0), φ(0), and ρ(0) are varied, the dynamical behaviors of the system will also change with different oscillations.

Fig. 15. Evolution process vs. u1(0) of co-existent oscillations: (a) chaotic oscillation with u1(0) = 0.01; (b) quasi chaotic oscillation with u1(0) = 2.0; (c) period-2 oscillation with u1(0) = 2.5; (d) period-1 oscillation with u1(0) = 5.0; (e) period-1 oscillation with u1(0) = 9.0; (f) stable equilibrium point with u1(0) = 9.9.
Table 1.

Initial values and corresponding states of Fig. 15.

.

The attractive basin, which changes with the system parameters, is an important tool to analyze co-existent attractors of circuits. Figure 16 gives the attractive basin of the proposed oscillation circuit in the cross section of φ = 0.01 and ρ = 0.01 when a = 1.0, b = 6.0, c = 4.5, d = 0.77, e = 4.0, and f = 1.0. It includes six different color regions, representing six different types of attractors in the given value regions of the two parameters. From the graph of the attractive basin, we can see that the oscillation state varies with the increases of u1(0) and u2(0). Several detailed co-existent attractors are given in Fig. 17. Figure 17(a) belongs to the type of co-existence of two different chaotic attractors. Figure 17(b) belongs to the type of chaotic attractor and limit cycle. Figure 17(c) belongs to the type of co-existence of two different limit cycles. Figure 17(d) belongs to the type of limit cycle and stable equilibrium points. Thus, figures 17(a) and 17(c) show homogeneous co-existent attractors, while figures 17(b) and 17(d) show inhomogeneous co-existent attractors. Table 2 lists the initial values and the corresponding types of co-existent attractors.

Fig. 16. Attractive basin on the cross section of φ = 0.01 and ρ = 0.01, with a = 1.0, b = 6.0, c = 4.5, d = 0.77, e = 4.0, and f = 1.0.
Fig. 17. (a) and (c) Homogeneous co-existent attractors; (b) and (d) inhomogeneous co-existent attractors.
Table 2.

Initial values and corresponding types of co-existent attractors of Fig. 17.

.
4. Dimension reduction analysis
4.1. Dimension reduction equations of the oscillator

For the purpose of reducing the complexity of solving Eq. (12), by integrating with respect to t at both sides of Eq. (11), and assuming all the initial values of the integrals are zero, we can change Eq. (11) to the following solvable 3-dimensional equations:

where is the flux of C1, is the flux of C2, and is the integral of the flux through the meminductor LM. By inserting into Eq. (13) and setting x = φ1, y = φ2, z = ρM, a = α/C1, b = β/(2C1), c = 1/(C1R), d =G/C2, e = α/C2, f = β/(2C2), equation (13) can be rewritten as

When a = 1.0, b = 20.0, c = 4.0, d = 0.8, e = 4.0, and f = 2.0, system (14) is a chaotic oscillator with the phase portraits shown in Fig. 18, where panel (d) is the relationship of ρ and q. Its Lyapunov exponents are computed as LE1 = 0.1106, LE2 = 0, and LE3 = −3.3106. The projection of the 3-dimensional Poincaré map onto the x–z plane is shown in Fig. 19(a), and the time domain waveforms of the three state variables are shown in Fig. 19(b).

Fig. 18. (a) Chaotic phase portraits on (a) x–y, (b) x–z, and (c) y–z planes; (d) relationship of ρ and q.
Fig. 19. (a) Poincaré map projected onto the x–z plane; (b) chaotic oscllation waveforms of the meminductor-based oscillator.
4.2. Dissipativeness and stability of the oscillator

The divergence of system (14) can be expressed as

If ∇V = −c+d < 0, the system is dissipative. Therefore, besides periodic oscillation, to make sure there will be chaotic oscillations included, the condition should be c > d.

Let ẋ = ẏ = ż = 0, we can obtain equilibrium points E1 (0,0,0) and , where z0 = (ad − ce)/(cf − bd). When system (14) is linearized at equilibrium point E1(0,0,0), we have the Jacobian matrix

The according characteristic equation is

By the Routh–Hurwitz rule, we conclude that if

are all positive, the eigenvalues of the system are all negative, and the system is stable. When the parameters are a = 1.0, b = 20.0, c = 4.0, d = 0.8, e = 4.0, and f = 2.0, we have λ1 = −3.7853 and λ23 = 0.2926±1.9824i. λ1 is a negative real number, and λ2 and λ3 are a pair of conjugate complex numbers with a positive real part, so E1 is an unstable saddle point.

When system (14) is linearized at equilibrium point E2, we have the Jacobian matrix

The corresponding characteristic equation is

where a1 = c − d, a2 = a + e + 2bz0 + 2f z0 − cd, and a3 = ce − ad + 2c f z0 − 2bdz0. When Δ1 = a1, Δ2 = (a1a2 − a3)/a1, and Δ3 = a3 are not all positive, it is possible to produce a chaotic oscillation. Let a = 1.0, b = 20.0, c = 4.0, d = 0.8, e = 4.0, and f = 2.0, the corresponding eigenvalues are λ1 = 0.1768 and λ23 = −1.6884±9.1185i. λ1 is a positive real root, λ2 and λ3 are a pair of conjugate complex roots with a negative real part, so it is also an unstable equilibrium point.

4.3. Dynamical properties with varying a

Let a = α/C1 in Eq. (14), where α = 0.03 is kept constant, then the value of a depends on the parameter C1. With varying a, the system will be in different states because of the change of the equilibrium point stability. When a is varied in the region [1, 2], the corresponding Lyapunov exponent spectrum is shown in Fig. 20(a) (all the values of LE3 are less than −3, for the purpose of clarity, the third curve of the Lyapunov exponent is not shown). The bifurcation diagram of x varying with a is illustrated in Fig. 20(b).

Fig. 20. (a) Lyapunov exponent spectrum with respect to a; (b) bifurcation diagram with respect to a.

From Fig. 20, we can see that the oscillator undergoes the process from the abruptly chaotic oscillation to periodic oscillation by an anti period-doubling bifurcation, then to chaotic oscillation again, and at last it goes to a stable periodic state via the repeated bifurcations from chaos to period. Note that when a = 1, a chaotic oscillation called abrupt chaos suddenly emerges.

4.4. Dynamical properties with varying d

In Eq. (14), by only varying d, the system will be in different states. When d is varied in the interval [0.1, 1], the resulted Lyapunov exponent spectrum and the corresponding bifurcation diagram of x are shown in Fig. 21.

Fig. 21. (a) Lyapunov exponent spectrum with respect to d; (b) bifurcation diagram with respect to d.

From Fig. 21, we can see that the system can generate multiple periodic oscillations and chaotic oscillations through a complex bifurcation process. When d ∈ [0.1,0.22], it is in the period-1 state; when d ∈ [0.22,0.78], it is in the period-doubling bifurcation process, and when d ∈ [0.78,1], it is in the chaotic state but with several periodic windows. The typical phase portraits with varying d are depicted in Fig. 22, where the complex bifurcation that changes from equilibrium point to period-1 and to period-2, then to period-4, and at last to chaos are observed clearly.

Fig. 22. (a) Equilibrium point with d = 0.17; (b) period-1 with d = 0.25; (c) period-1 with d = 0.7; (d) period-2 with d = 0.73; (e) period-4 with d = 0.75; (f) chaos with d = 0.8.
4.5. Dynamical map with varying c and d

The bifurcation parameters c and d impact the dynamical properties of the system. The dynamical map can present the impact on the dynamical properties of the system, from which we can obtain the dynamical state with detailed parameter ranges. Figure 23 gives the two-dimensional dynamical map with respect to c and d, in the meantime, it describes the dependent relationship between the dynamical behaviors and the bifurcation parameters. In the dynamical map, we set a = 1.0, b = 20.0, e = 4.0, and f = 2.0.

Fig. 23. Dynamical map with respect to bifurcation parameters c and d.

In Fig. 23, the blue area marked C represents that the system is in the chaotic state; the yellow area marked P represents that the system is in the periodic state; while the red area marked S stands for the stable state. This dynamical map depicts the impact of the parameters on the dynamical behavior of the system visually. When d ∈ [0.1,0.2], the system converges to point attractors; when d ∈ [0.2,0.8], the system runs at periodic orbits; while when d ∈ [0.8,1], the system is in chaotic states.

By inserting a = 1.0, b = 20.0, e = 4.0, and f = 2.0 into Eq. (14), and linearizing it around equilibrium point E1(0,0,0), we obtain the characteristic equation

By the Routh–Hurwitz rule, we conclude that when cd(c − d) + 4d − c < 0, equilibrium point E1 is stable and this system will converge into point attractors (as shown in Fig. 22(a)). The bifurcation border line between the red region and the yellow region at the bottom of Fig. 23 is determined by

The stable equilibrium points stand in the area below the line, because the stable area must satisfy cd(c − d) + 4d − c < 0.

4.6. Dynamical map with varying a and e

Figure 24 gives the dynamical map defined by a and e, where we set the other parameters to be b = 20.0, c = 4.36, d =0.97, and f = 2.0. There are two different areas with different colors in Fig. 24, where the blue area marked C and the yellow area marked P represent chaotic and periodic states, respectively. Thus when a is small, the system is in the chaotic state; as a gets large, it transfers from chaotic orbit to period orbit.

Fig. 24. Dynamical map defined by bifurcation parameters a and e.
5. Realization of the meminductor-based oscillator

By combining Eq. (14) and the equivalent circuit model of the meminductor in Fig. 3, we design an equivalent circuit of the meminductor-based oscillator and verify the circuit by experiments. The designed circuit is shown in Fig. 25 and the according simulation results by using Multisim are shown in Fig. 26.

Fig. 25. Equivalent circuit of the meminductor-based chaotic oscillator.
Fig. 26. (a) Phase obits of (a) x–y, (b) x–z, (c) y–z for the meminductor-based oscillator.

The equivalent circuit is designed based on the state equation (14). In Fig. 25, U1–U8 are operational amplifiers used for implementing the operations of inverse proportion, integration, and add for variables x, y, and z; and A1 and A2 are multipliers used for implementing z2. According to Fig. 25, the state equations can be described as

We apply time and scale transformations for Eq. (17), and let t = KT, x = MX, y = MY, z = MZ, where K = 1000 and M = 2 are the scale transformation factors. Then equation (1) is transformed to

Comparing Eqs. (18) and (14), we have

When C1 = C2 = 10 nF, R1 = R6 = R9 = 2.5 kΩ, R2 = R4 = R7 = R11 = R12 = R14 = R15 = R16 = R17 = R18 = 10 kΩ, R3 = 250 Ω, R5 = R8 = R13 = 100 kΩ, and R10 = 12.5 kΩ, the corresponding parameters are a = 1.0, b = 20.0, c = 4.0, d = 0.8, e = 4.0, and f = 2.0. The simulated phase orbits for the circuit are shown in Fig. 26.

The experimental meminductor-based oscillator obtained according to Fig. 25 is depicted in Fig. 27, where R1 and R10 are variable resistors. We can change the resistances of R1 and R10 to change the state of the output of the oscillator. When R1 = 2.86 kΩ and the other parameters are the same as above, the oscillator can produce a double-period signal, the corresponding oscilloscope displays are shown in Figs. 28(a), 28(c), and 28(e) (oscilloscope DSO1302A was used). When R1 = 2.5 kΩ, the oscillator will produce chaotic obits, which are shown in Figs. 28(b), 28(d), and 28(f). The dynamical properties of the circuit can be changed by adjusting the value of R1. Figure 29 demonstrates a series of phase portraits of y–z in different states. Figure 30 demonstrates the experimental setup.

Fig. 27. Experimental circuit board of the meminductor-based oscillator.
Fig. 28. (a) Experimental periodic orbit of x–y; (b) experimental chaotic orbit x–y; (c) experimental periodic orbit of x–z; (d) experimental chaotic orbit of x–z; (e) experimental periodic orbit of y–z; (f) experimental chaotic orbit of y–z.
Fig. 29. Experimentally observed bifurcations: (a) experimental period-1 orbit with R1 = 3.3 kΩ; (b) experimental period-2 orbit with R1 = 2.85 kΩ; (c) experimental period-4 orbit with R1 = 2.81 kΩ; (d) experimental chaotic orbit with R1 = 2.78 kΩ.
Fig. 30. (a) Experimental setup of meminductor-based oscillator; (b) experimental setup of the meminductor model.
6. Conclusion

A mathematical model of the flux-controlled meminductor and its equivalent circuit models of φ–i and ρ–q have been presented. In the circumstance that a physical meminductor has not been realized, we can use such models instead of the real meminductor for studying the properties and applications of meminductors theoretically and experimentally. On the other hand, based on the model, we have designed and built a memintuctor-based oscillator, which exhibits some complex behaviors, such as co-existent oscillations, abrupt chaos, period-doubling bifurcation, inverse period-doubling bifurcation, and broader chaotic attractive field. Especially, it is sensitive to the initial conditions which could be used to produce co-existent oscillations and to generate consecutive and stable pseudo random sequences as multi sources, which can be applied in information encryption and secret communications.

Reference
1Chua L O 1971 IEEE Trans. Circuit Theory 18 507
2Strukov D BSnider G SStewart D RWilliams R S 2008 Nature 453 80
3Chua L O1978Guest Lectures of the 1978 European Conference on Circuit Theory and Design81
4Chua L O 2003 Proc. IEEE 91 1830
5Di Ventra MPershin Y VChua L O 2009 Proc. IEEE 97 1717
6Chua L O2009IEEE Expert Now SHORT Course[Online] Available at: http://ieeexplore.ieee.org/xpl/modulesabstract.jsp?mdnumber=EW1091
7Jung C MChoi J MMin K S 2012 IEEE Trans. Nanotechnol. 11 611
8Eshraghian KCho K RKavehei OKang S KAbbott DSteve S M 2011 IEEE Trans. Vlsi Systa. 19 1407
9Bao B CLiu ZXu J P 2010 Chin. Phys. 19 030510
10Bao B CHu F WLiu ZXu J P 2014 Chin. Phys. 23 070503
11Xu B R 2013 Acta Phys. Sin. 62 190506 (in Chinese)
12Fouda M ERadwan A G 2014 Microelectron 45 1372
13Pershin Y VDi Ventra M 2011 Adv. Phys. 60 145
14Biolek DBiolek ZBiolkova V 2011 Circ. Sig. Process. 66 129
15Liang YYu D SChen H2013Acta Phys. Sin.62158501(in Chinese)
16Abdelouahab M SLozi RChua L O 2014 Int. J. Bifurcation Chaos 24 1430023
17Pershin Y VDi Ventra M 2010 Electron. Lett. 46 517
18Shi Z YWang C LBao B CFeng F 2014 UEST 43 845
19Sah M PBudhathoki R KYang CKim H 2014 Circuit ISCAS 33 2363
20Pershin Y VDi Ventra M 2011 Adv. Phys. 60 145
21Kim K MJeong D SHwang C S 2011 Nanotechnology 22 254002
22Muthuswamy BChua L 2010 Int. J. Bifurcation Chaos 20 1567
23Sprott J CWang XChen G 2013 Int. J. Bifurcation Chaos 23 1350093
24Molaie MJafari SSprott J CHashemi S M R H 2013 Int. J. Bifurcation Chaos 23 1350188