† Corresponding author. E-mail:
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).
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.
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.
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:
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. (
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:
![](cpb_25_9_090502/cpb160481ieqn1.gif)
![](cpb_25_9_090502/cpb160481ieqn1.gif)
![](cpb_25_9_090502/cpb160481ieqn2.gif)
Let α = 0.03, β = 0.36, and φ(t) = Φm sin(2πFt), according to Eq. (
![](cpb_25_9_090502/cpb160481ieqn3.gif)
![](cpb_25_9_090502/cpb160481ieqn4.gif)
The time domain waveforms of the current and the flux are shown in Fig.
An equivalent circuit is important for the meminductor research since a physical meminductor has not been realized. The equivalent circuit model based on Eq. (
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
From Figs.
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.
Figures
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.
The equivalent circuit model of the ρ–q property for the meminductor based on Eq. (
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
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.
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
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.
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. (
![]() | Fig. 14. (a) Lyapunov exponent spectrum with initial value u1 (0); (b) the corresponding bifurcation diagram. |
From Fig.
Figure
![]() | Table 1. Initial values and corresponding states of Fig. |
The attractive basin, which changes with the system parameters, is an important tool to analyze co-existent attractors of circuits. Figure
![]() | 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. |
For the purpose of reducing the complexity of solving Eq. (
![](cpb_25_9_090502/cpb160481ieqn7.gif)
![](cpb_25_9_090502/cpb160481ieqn8.gif)
![](cpb_25_9_090502/cpb160481ieqn9.gif)
![](cpb_25_9_090502/cpb160481ieqn10.gif)
When a = 1.0, b = 20.0, c = 4.0, d = 0.8, e = 4.0, and f = 2.0, system (
![]() | Fig. 18. (a) Chaotic phase portraits on (a) x–y, (b) x–z, and (c) y–z planes; (d) relationship of ρ and q. |
The divergence of system (
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
When system (
Let a = α/C1 in Eq. (
![]() | Fig. 20. (a) Lyapunov exponent spectrum with respect to a; (b) bifurcation diagram with respect to a. |
From Fig.
In Eq. (
![]() | Fig. 21. (a) Lyapunov exponent spectrum with respect to d; (b) bifurcation diagram with respect to d. |
From Fig.
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
In Fig.
By inserting a = 1.0, b = 20.0, e = 4.0, and f = 2.0 into Eq. (
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.
Figure
By combining Eq. (
The equivalent circuit is designed based on the state equation (
We apply time and scale transformations for Eq. (
The experimental meminductor-based oscillator obtained according to Fig.
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.
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