1 Introduction

Memristor Circuits for Simulating Nonlinear Dynamics and Their Periodic Forcing
Makoto Itoh111After retirement from Fukuoka Institute of Technology, he has continued to study the nonlinear dynamics on memristors.
1-19-20-203, Arae, Jonan-ku,
Fukuoka, 814-0101 JAPAN
Email: itoh-makoto@jcom.home.ne.jp

In this paper, we show that the dynamics of a wide variety of nonlinear systems such as engineering, physical, chemical, biological, and ecological systems, can be simulated or modeled by the dynamics of memristor circuits. It has the advantage that we can apply nonlinear circuit theory to analyze the dynamics of memristor circuits. Applying an external source to these memristor circuits, they exhibit complex behavior, such as chaos and non-periodic oscillation. If the memristor circuits have an integral invariant, they can exhibit quasi-periodic or non-periodic behavior by the sinusoidal forcing. Their behavior greatly depends on the initial conditions, the parameters, and the maximum step size of the numerical integration. Furthermore, an overflow is likely to occur due to the numerical instability in long-time simulations. In order to generate a non-periodic oscillation, we have to choose the initial conditions, the parameters, and the maximum step size, carefully. We also show that we can reconstruct chaotic attractors by using the terminal voltage and current of the memristor. Furthermore, in many memristor circuits, the active memristor switches between passive and active modes of operation, depending on its terminal voltage. We can measure its complexity order by defining the binary coding for the operation modes. By using this coding, we show that in the forced memristor Toda lattice equations, the memristor’s operation modes exhibit the higher complexity. Furthermore, in the memristor Chua circuit, the memristor has the special operation modes.
Keywords:  Keywords: memristor; chaos; quasi-periodic; non-periodic; numerical instability; integral invariant; attractor reconstruction; passive; active; instantaneous power; complexity order; memristor’s operation modes; Chua circuit; Van der Pol oscillator; Hamilton’s equations; Hamiltonian; Toda lattice equations; Lotka-Volterra equations; ecological predator-prey model; Rössler equations; Lorenz equations; Brusselator equations; Gierer-Meinhardt equations; Tyson-Kauffman equations; Oregonator equations; sine-Gordon equation; tennis racket equations; pendulum equations; laser model.

## 1 Introduction

The dynamics of -dimensional autonomous systems can be transformed into the dynamics of two-element extended memristor circuits. The internal state of the memristors in these two-element circuits have the same dynamics as -dimensional autonomous systems [1]. Thus, the memristors are essential dynamical elements needed in the modeling of complex nonlinear dynamical phenomena. In this paper, based on the above research results, we show that the dynamics of a wide variety of nonlinear systems, not only in physical and engineering systems, but also in biological and chemical systems and, even, in ecological systems, can be simulated or modeled by the dynamics of memristor circuits. It has the advantage that we can apply nonlinear circuit theory to analyze the dynamics of memristor circuits.

It is known that the dynamics of Chua’s circuit and Van der Pol oscillator can be realized by using an ideal active memristor and some linear elements [2]. However, almost nonlinear systems can not satisfy the circuits equations without change. Thus, in order to transform their nonlinear equations into the memristor circuit equations, we use two methods, one is the exponential coordinate transformation, and the other is the time-scaling change [1, 3, 4]. The resulting memristor circuits have the same dynamics as the nonlinear systems. Furthermore, by connecting an external periodic forcing to these memristor circuits, they can exhibit complex behavior, such as chaos and non-periodic oscillation. If the memristor circuits have an integral invariant, then they can exhibit quasi-periodic or non-periodic behavior, which greatly depends on the initial conditions, the circuit parameters, and the maximum step size of the numerical integration. Furthermore, an overflow (outside the range of data) is likely to occur due to the numerical instability in long-time simulations. Thus, in order to generate a non-periodic oscillation, we have to choose the initial conditions, the parameters, and the maximum step size, carefully. Furthermore, noise may considerably affect the behavior of physical circuits.

We also show that if we plot the terminal voltage against current of the memristor in the circuits, we can get the reconstruction of chaotic attractor on the two-dimensional plane. Furthermore, if we plot the instantaneous power versus the terminal voltage of the active memristor, then the locus lies in the first and the third quadrants, and it is pinched at the origin in many memristor circuits. It looks exactly like the loci of the passive memristor when a periodic source is supplied. Thus, the active memristor switches between passive and active modes of operation depending on its terminal voltage. However, in the forced memristor Toda lattice equations, the locus exhibits more complicated behavior, that is, it switches between four modes of operation. In order to measure the complexity order, we define the binary coding for the above memristor’s operation modes. By using this coding, we show that in the forced memristor Toda lattice equations, the memristor’s operation modes exhibit the higher complexity. Furthermore, in the memristor Chua circuit, the active memristor exhibits the special operation modes, which is quite different from the other memristor circuits.

## 2 Three-element Memristor Circuit

Let us consider the three-element memristor circuit in Figure 1, which consists of an inductor , a battery , and a current-controlled extended memristor.

The terminal voltage and the terminal current of the current-controlled extended memristor are described by

V-I characteristics of the extended memristor (1)

Here, , is a continuous scalar-valued function,
and (see Appendix A).

The dynamics of the above three-element memristor circuit is given by

Three-element memristor circuit equations (2) where denotes the inductance of the inductor, denotes the voltage of the battery, and .

Assume that and . Then Eq. (2) can be recast into the form

Three-element memristor circuit equations with and (3) where and .

### 2.1 Brusselator equations

The Brusselator is a theoretical model for a type of autocatalytic reaction. The dynamics of the Brusselator is given by

Brusselator equations (4) where and are constants.

Consider the three-element memristor circuit in Figure 1 with . Then the dynamics of this circuit is given by Eq. (3). Assume that Eq. (3) satisfies

 E=A,^R(x,i)=−{ix−(B+1)},~f1(x,i)=Bi−i2x. (5)

Then we obtain

Memristor Brusselator equations (6) where and are constants.

Equations (4) and (6) are equivalent if we change the variables

 i=u, x=v. (7)

The terminal voltage and the terminal current of the current-controlled extended memristor in Figure 1 are given by

V-I characteristics of the extended memristor (8) where and .

It follows that the Brusselator equations (4) can be realized by the three-element memristor circuit in Figure 1. Equations (4) and (6) exhibit periodic oscillation (limit cycle). When an external source is added as shown in Figure 2, the forced memristor Brusselator equations can exhibit chaotic oscillation [5]. The dynamics of this circuit is given by

Forced memristor Brusselator equations (9) where and are constants.

We show the chaotic attractor, Poincaré map, and locus of Eq. (9) in Figures 3, 4, and 5(a), respectively. The following parameters are used in our computer simulations:

 A=E=0.4, B=1.2, r=0.05, ω=0.81. (10)

The locus moves in the first quadrant, that is, it moves in the passive region, since the instantaneous power of the extended memristoris positive, that is,

 PM(t)\lx@stackrel△=iM(t)vM(t)>0. (11)

Hence, the instantaneous power is dissipated in the extended memristor, which is delivered from the forcing signal and the inductor. Furthermore, the locus is not pinched at the origin as shown in Figure 5(a), since the trajectory does not tend to the origin.

We define next the instantaneous power of the two circuit elements, that is, the instantaneous power of the extended memristor and the battery by

 pME(t)\lx@stackrel△=iM(t)vME(t), (12)

where , and denotes the voltage of the battery. That is, denotes the voltage across the extended memristor and the battery. We show the locus in Figure 5(b). Observe that the locus is pinched at the origin, and it lies in the first and the third quadrants. Thus, the instantaneous power delivered from the forced signal and the inductor is dissipated when . However, the instantaneous power is not dissipated when . We conclude as follow:

Behavior of the extended memristor   Assume that Eq. (9) exhibits chaotic oscillation. Then, we obtain the following results: The extended memristor defined by Eq. (8) is operated as a passive element. The instantaneous power of the memristor is dissipated in this extended memristor, which is delivered from the forcing signal and the inductor. When , the instantaneous power of the extended memristor and the battery is not dissipated. However, when , the instantaneous power is dissipated.

Note that in Eq. (8) is the internal state of the extended memristor. Thus, we might not be able to observe it. However, we can reconstruct the chaotic attractor into two dimensional Euclidean space (plane) by using

 (i(t), i′(t)), (13)

where (see [6] for more details). Furthermore, the locus in Figure 5(a) is considered to be the reconstruction of the chaotic attractor on the two-dimensional plane, since

 (iM(t),vM(t))≡(i(t), −i′(t)+A+rsin(ωt)), (14)

where . We show their trajectories and Poincaré maps in Figures 6 and 7, respectively. We can also reconstruct the chaotic attractor into the three-dimensional Euclidean space by using

 (i(t), i′(t), i′′(t)), (15)

or

 (iM(t),vM(t),i′′M(t))≡(i(t), −i′(t)+A+rsin(ωt),i′′(t)), (16)

where . We show the reconstructed three-dimensional attractors in Figure 8. We can apply the above reconstruction methods to other examples in this paper.

### 2.2 Diffusion-less Gierer-Meinhardt equations

Diffusion-less Gierer-Meinhardt equations [7, 8, 9] is defined by

Diffusion-less Gierer-Meinhardt equations (17) where and are constants.

Let us consider the three-element memristor circuit in Figure 1. The dynamics of this circuit given by Eq. (3). Assume that Eq. (3) satisfies

 E=0,^R(x,i)=−(ix−b),~f1(x,i)=i2−cx. (18)

Then we obtain

Memristor diffusion-less Gierer-Meinhardt equations (19) where and are constants.

Equations (17) and (19) are equivalent if we change the variables

 i=u, x=v. (20)

The terminal voltage and the terminal current of the current-controlled extended memristor in Figure 1 are given by

V-I characteristics of the extended memristor (21) where .

The above small-signal memristance satisfies

 limx→0|^R(x,iM)|=∣∣∣−(iMx−b)∣∣∣=∞, (22)

when . In order to avoid this singularity, we use the different time-scaling [10]. That is, after time scaling by , Eqs. (17), (19), and (21) assume the equivalent forms

Diffusion-less Gierer-Meinhardt equations with time scaling (23) where and are constants,          Memristor diffusion-less Gierer-Meinhardt equations with time scaling (24) where and are constants,

and

V-I characteristics of the extended memristor (25) where ,

respectively. Similarly, Eq. (24) can be realized by the three-element memristor circuit in Figure 1, where

 E=0,^R(x,i)=−(i−bx),~f1(x,i)=(i2−cx)x, (26)

Note that the above time scaling maps orbits between systems (17) and (23) in a one-to-one manner except at the singularity , although it may not preserve the time orientation of orbits.

Equations (17) and (24) exhibit periodic oscillation (limit cycle). When an external source is added as shown in Figure 2, the forced memristor circuit can exhibit chaotic oscillation. The dynamics of this circuit is given by

Forced memristor diffusion-less Gierer-Meinhardt equations with time scaling (27) where and are constants.

We show the chaotic attractor, Poincaré map, and locus of Eq. (27) in Figures 9, 10, and 11, respectively. The following parameters are used in our computer simulations:

 b=0.65, c=0.796, r=0.2, ω=0.5. (28)

The locus in Figure 11 lies in the first and the fourth quadrants. Thus, the extended memristor defined by Eq. (25) is an active element. Let us next consider an instantaneous power defined by . Then we obtain the locus in Figure 12. Observe that the locus is pinched at the origin, and the locus lies in the first and the third quadrants. Thus, when , the instantaneous power delivered from the forced signal and the inductor is dissipated in the memristor. However, when , the instantaneous power delivered from the forced signal and the inductor is not dissipated in the memristor. Note that the locus in Figure 12 looks exactly like the locus of the “passive” memristor, whose locus lies in the first and the third quadrants [12]. Hence, the memristor switches between passive and active modes of operation, depending on its terminal voltage. We conclude as follow:

Switching behavior of the memristor   Assume that Eq. (27) exhibits chaotic oscillation. Then the extended memristor defined by Eq. (25) can switch between “passive” and “active” modes of operation, depending on its terminal voltage.

In order to obtain the results shown in Figures 9-12, we have to choose the initial conditions carefully. It is due to the fact that a periodic orbit (drawn in magenta)222Without loss of generality, we can use the terminology “periodic orbit” in order to describe a “periodic trajectory” of the nonautonomous systems, such as Eqs. (9) and (27) (see “Duffing’s Equation” in Sec. 2.2 of [11])). coexists with a chaotic attractor (drawn in blue) as shown in Figure 13.

As stated in Sec. 2.1, we can reconstruct the chaotic attractor into two dimensional plane by using

 (i(t), i′(t)). (29)

Furthermore, the locus in Figure 11 is considered to be the reconstruction of the chaotic attractor on the two-dimensional plane, since

 (iM(t),vM(t))≡(i(t), −i′(t)+rsin(ωt)), (30)

where . We show their trajectories and Poincaré maps in Figures 14 and 15, respectively. We can also reconstruct the chaotic attractor into the three-dimensional Euclidean space by using

 (i(t), i′(t) i′′(t)), (31)

or

 (32)

We show the reconstructed three-dimensional attractors in Figure 16.

### 2.3 Tyson-Kauffman equations

The dynamics of the Tyson-Kauffman equations [13] can be described by

Tyson-Kauffman equations (33) where and are constants.

Consider the three-element memristor circuit in Figure 1. The dynamics of this circuit given by Eq. (3). Assume that Eq. (3) satisfies

 E=A,^R(x,i)=(B+x2),~f1(x,i)=Bi+ix2−x. (34)

Then we obtain

Memristor Tyson-Kauffman equations (35) where and are constants

Equations (33) and (35) are equivalent if we change the variables

 i=u, x=v. (36)

In this case, the extended memristor in Figure 1 is replaced by the generic memristor (see Appendix A). That is,

 ^R(x,i)=~R(x)=(B+x2). (37)

The terminal voltage and the terminal current of the current-controlled generic memristor are described by

V-I characteristics of the generic memristor (38) where .

It follows that the Tyson-Kauffman equations (33) can be realized by the three-element memristor circuit in Figure 1. Equations (33) and (35) can exhibit periodic oscillation (limit cycle). When an external source is added as shown in Figure 2, the forced memristor Tyson-Kauffman equations can exhibit chaotic oscillation. The dynamics of the circuit is given by

Forced memristor Tyson-Kauffman equations (39) where and are constants.

We show the chaotic attractors, Poincaré maps, and loci of Eq. (39) in Figures 17, 18, and 19, respectively. In our computer simulations, we used the following two kinds of the parameters:

 (a)A=0.5,B=0.00803,C=0.01,r=0.5,ω=0.5,} (40)

and

 (b)A=0.5,B=0.0079,C=0.01,r=0.55,ω=0.5.} (41)

Note that the locus in Figure 19(a) moves in the first quadrant, and the locus in Figure 19(b) moves in the first and third quadrants. That is, they move in the passive region, since the instantaneous power defined by

 pM(t)\lx@stackrel△=iM(t)vM(t), (42)

is not negative. In this case, the power is dissipated in the generic memristor, which is delivered from the forcing signal and the inductor.

Let us define next the instantaneous power of the two circuit elements, as stated in Sec. 2.1. That is, we define the instantaneous power of the extended memristor and the battery by

 pME(t)\lx@stackrel△=iM(t)vME(t), (43)

where , and denotes the voltage of the battery. That is, denotes the voltage across the extended memristor and the battery. We show the locus in Figure 20. Observe that the locus is pinched at the origin, and it lies in the first and the third quadrants. Thus, the instantaneous power delivered from the forced signal and the inductor is dissipated when . However, the instantaneous power is not dissipated when . Thus, we conclude as follow:

Behavior of the generic memristor   Assume that Eq. (39) exhibits chaotic oscillation. Then, we obtain the following results: The generic memristor defined by Eq. (38) is operated as a passive element. If we define the instantaneous power of the generic memristor by , then is dissipated in this generic memristor, which is delivered from the forcing signal and the inductor. If we define the instantaneous power of the two elements, that is, the instantaneous power of the generic memristor and the battery, by , then is not dissipated when . However, is dissipated when .

As stated in Sec. 2.1, we can reconstruct the chaotic attractor into two dimensional plane by using

 (i(t), i′(t)). (44)

Furthermore, the loci in Figure 19 are considered to be the reconstruction of the chaotic attractor on the two-dimensional plane, since

 (iM(t),vM(t))≡(i(t), −i′(t)+rsin(ωt)), (45)

where . We show their trajectories and Poincaré maps in Figures 21 and 22, respectively. We can also reconstruct the chaotic attractor into the three-dimensional Euclidean space by using

 (i(t), i′(t) i′′(t)), (46)

or

 (47)

We show the reconstructed three-dimensional attractors in Figure 23.

### 2.4 Lotka-Volterra equations

Consider Hamilton’s Equations defined by

Hamilton’s equations (48) where and denote the coordinate and the momentum and is the Hamiltonian.

Let us define the Hamiltonian:

Hamiltonian (49) where are constants.

From Eq. (48), we obtain

 dqdt=∂H∂p=−b+ap,dpdt=−∂H∂q=c−dq.\vspace2mm (50)

After time scaling by , we obtain the associated Pfaff’s equation [10]

 dqdτ=(−b+ap)pq,dpdτ=(c−dq)pq.\vspace2mm (51)

Equation (51) can be recast into the Lotka-Volterra equations [14]

Lotka-Volterra equations (52) where are constants.

Equation (52) has the Hamiltonian (49) as its integral invariant, that is,

 dH(q,p)dτ=0. (53)

Consider next the three-element memristor circuit in Figure 1. The dynamics of this circuit given by Eq. (3). Assume that Eq. (3) satisfies

 E=0,^R(x,i)=−(cx−d),~f1(x,i)=(a−bi)x. (54)

Then we obtain

Memristor Lotka-Volterra equations (55) where are constants.

Equations (52) and (55) are equivalent if we change the variables

 i=p, x=q. (56)

In this case, the extended memristor in Figure 1 is replaced by the generic memristor (see Appendix A). That is,

 ^R(x,i)=~R(x)=−(cx−d). (57)

Thus, the terminal voltage and the terminal current of the current-controlled generic memristor are given by

V-I characteristics of the generic memristors (58) where .

It follows that the Lotka-Volterra equations (55) can be realized by the three-element memristor circuit in Figure 1.

The Lotka-Volterra equations (55) can exhibit a periodic orbit (one-dimensional curve), since they have

Integral   (59)

as its integral invariant. When an external source is added as shown in Figure 2, the forced Lotka-Volterra equations can exhibit a quasi-periodic or a non-periodic response,333In this paper, we use the terminology “non-periodic response” in order to describe “chaotic-like but non-attracting response”. which depends on initial conditions. The dynamics of the circuit is given by

Forced memristor Lotka-Volterra equations (60) where and are constants.

We show their non-periodic and quasi-periodic responses, Poincaré maps, and loci in Figures 24, 25(a), and 26, respectively. The following parameters are used in our computer simulations:

 a=2/3, b=4/3, c=1, d=1,r=0.04, ω=1, or 1.01. (61)

The loci in Figure 26 lie in the first and the fourth quadrants. Thus, the generic memristor defined by Eq. (58) is an active element. Let us next show the locus in Figure 27, where is an instantaneous power defined by . Observe that the locus is pinched at the origin, and the locus lies in the first and the third quadrants. Thus, when , the instantaneous power delivered from the forced signal and the inductor is dissipated in the generic memristor. However, when , the instantaneous power delivered from the forced signal and the inductor is not dissipated in the generic memristor. Hence, the memristor switches between passive and active modes of operation, depending on its terminal voltage. We conclude as follow:

Switching behavior of the memristor   Assume that Eq. (60) exhibits non-periodic response. Then the generic memristor defined by Eq. (58) can switch between “passive” and “active” modes of operation, depending on its terminal voltage.