Chaotic dynamics in memristive circuits

Memristors (abbreviation of memory and resistor), introduced as the fourth fundamental electrical circuit element, remember the electric charge that flowed through it. These nonlinear elements are considered as a class of two terminal resistive devices and offers a lot of possible applications in various areas. In this article, we review the dynamical behaviour of some electrical circuits with memristors. Initially, we show that a simple circuit with a capacitor and an inductor connected to a memristor exhibits periodic and chaotic attractors. After that, we show that the known Chua circuit, with a nonlinear resistance, can generate bifurcations and chaos. Substituting the nonlinear resistance by a memristor, the modified Chua circuit exhibits coexistence of attractors. Another known circuit, the Colpitts, is made of a combination of capacitors and inductors containing a bipolar junction transistor. We show that the modified Colpitts circuit, created by substituting the transistor by a memristor, presents multistability and coexistence of many attractors.


Introduction
In 1971, Chua [1] proposed the fourth fundamental passive circuit element, called memristor.Memristor is a portmanteau of the words memory and resistor.Its valuable property is the possibility of changing the resistance as a function of the current, memorising the alterations.The first practical memristive device was based on a thin of titanium dioxide and fabricated by a group at HP Research Labs in 2008 [2].It exhibits a pinched hysteresis between the current and the voltage.Nowadays other kinds of memristors have been developed [3].
Memristors offer a lot of potential applications in different areas, such as electrical engineering and computer science.They have been widely used in hardware security [4] and neural network [5].Wang et al. [6] proposed a memristive chaotic system to achieve the encryption of image.Tan and Wang [7] demonstrated that a neural model composed of two neurons based on the memristor can show firing pattern transitions and different attractors.The memristors can also be used to * Correspondence email address: kiarosz@gmail.comgenerate digital modulate signals [8] and to implement programmable circuits [9].
A great deal of research has been devoted to analyse the dynamic behaviour of circuits with memristors [10].Properties of basic electric circuits constructed from resistors, capacitors, inductors, and mainly memristors have been reported [11].
In this article, we introduce some chaotic electric circuits and how their dynamics is modified by including a memristor.Initially, we introduce a simple circuit, with a capacitor and an inductor coupled with a nonlinear active memristor [12], and show that this circuit presents periodic and chaotic attractors.After that, we present the common Chua and Colpitts nonlinear circuits that also exhibit chaotic attractors.The Chua circuit has been used as source of pseudo random signals and in secure communication [13].The Colpitts circuit is able to generate high frequency sinusoidal signals and optical signals with application in optoelectronics [14].However, when we replace the nonlinear resistance by a memristor in the Chua circuit, the modified circuit presents coexisting periodic and chaotic attractors [15].Furthermore, for a sinusoidal voltage stimulus, this modified Chua circuit presents complex transient dynamics [16].Finally, we introduce the Colpitts circuit composed of a combination of capacitors and inductors [17].Furthermore, when we substitute the exponential nonlinear term in the Colpitts system by an ideal memristor, we find, depending on the initial condition, extreme multistability [18].This paper is organised as follows.In Section 2, we introduce the memristor and a mathematical model, as well as some values of the parameters commonly used in the literature.A three circuit elements in series, a modified memristor Chua circuit, and a modified memristor Colpitts circuit are presented in Section 3. Depending on the parameter values, the three circuit elements in series can exhibit periodic and chaotic attractors, as shown in Subsection 3.1.Subsections 3.2 and 3.3 show that coexisting attractors can be find in modified memristor circuits based on Chua and Colpitts oscillators.Finally, we draw our conclusions in Section 4.

Memristor
Memristor is an abbreviation of memory resistor, namely a resistor with the capability to hold data in memory.It is composed of electrode and dielectric layers, as well as substrate.The materials play an important role in its performance [3].The memristor model reported by Strukov et al. [2] is based on a two-layer film composed of undoped (TiO 2 ) and doped oxygen vacancies (TiO 2−x ) with platinum electrodes (Pt), as shown in Fig. 1.
The memristor model is described by [2] V = M • I, (1) where V is the voltage potential, M is the generalised resistance, I is the electric current, and w is the state variable of the device [19].R ON and R OFF correspond to the low and high resistance, respectively [11].D represents the full length of the memristor.Equation ( 1) can be rewritten as I = G • V , where the memductance is G = 1/M [10].The time evolution of w is given by  where η corresponds to the polarity, v 0 the voltage amplitude, T the period of potential oscillation, and t the time.In Eq. ( 3), f (w, p) = 1 − (2w − 1) 2p is the window function of the nonlinear dopant drift and p is a positive integer number [2,20,21].Table 1 shows the description and some values of the parameters commonly used in the literature [2].
To illustrate the typical memristor temporal behaviour, we apply a periodic potential V (t) to it.Thus, Fig. 2 displays the memresistance variable M , and current passing behind set I. The state variable of the device has the initial value w(t = 0) = w 0 as the minimal, as well as the initial resistance as maximal M (w 0 ).We see that oscillation in the applied potential (Fig. 2(a)) is able to change the state of the memristor device (Fig. 2(b)).As a consequence, the memresistance (Fig. 2(c)) generates oscillations in the current passing by the device (Fig. (2(d)).
Figure 3 exhibits I as a function of V for different values of w 0 .For w 0 = 0.0 nm, I is directly proportional to V , as shown in Fig. 3(a).In Figs.3(b) and 3(c) for w 0 = 0.5 nm and w 0 = 0.6 nm, respectively, we see a We consider the parameters according to Table 1.We observe periodic oscillations.current-voltage characteristic of the memristor device, that is the pinched hysteresis effect.For w 0 = 0.8 nm, the I − V curve is a straight line through the origin.To obtain the I ×V curves, we consider V = v 0 •sin(2πt/T ).

Three circuit elements in series
The nonlinear dynamic behaviour of memristors have been analysed in oscillatory and chaotic circuits [22].Muthuswamy and Chua [12] proposed a circuit with a linear passive inductor, a linear passive capacitor, and a nonlinear active memristor.Figure 4 displays a schematic circuit diagram that has the three elements in series.
The dynamic equations of the circuit are given by ẋ = y C ,   where x corresponds to the voltage across the capacitor C and y is related to the current through the inductor L. The variable z is the internal state of the memristive system.The parameters α and β are associated with the memristive device.

Modified memristor Chua circuit
Around 1983, Chua invented an electronic circuit that produces irregular behaviour [25].The circuit has a nonlinear negative resistance, known as Chua diode.In 1984, Matsumoto [26] demonstrated numerically the appearance of chaotic attractors in the Chua circuit.Experiments confirming the existence of periodicity and chaos from the Chua circuit were reported by Zhong and Ayrom [27] in 1985.After that, dynamical properties of this circuit have been theorethically investigated [28,29].
A memristive circuit can be derived from the Chua circuit by replacing the Chua diode with the memristor [22], as shown in Fig. 7. Periodic and chaotic trajectories were observed in experimental setup of modified Chua circuit [30].Xu et al. [31] found multiple attractors in a memristive Chua circuit.
The dimensionless model of the modified Chua circuit is described by [15] ẋ = a(y    Depending on the initial conditions, the trajectory can move toward different attractors.Figure 10(a) displays initial conditions x(0) and y(0) for orbits converging to three different attractors with period one (white, blue, and red).In Fig. 10(b), we observe trajectories going to a chaotic attractor (green region) and others going to a attractor with period one (red region).

Modified memristor Colpitts circuit
In 1918, Colpitts [32] invented a type of oscillatory circuit that utilises a combination of capacitors and inductors.The oscillating frequency can be adjusted from a few Hertz up to 10 9 Hertz (microwave region).For this circuit, there are numerical and experimental evidences of periodic and chaotic behaviours [33][34][35].[36].
Considering normalised parameters and dimensionless state variables, the mathematical model is given by [37] where x and y are associated with the voltages across the capacitors, z is related to the current through the inductor, while g, Q, and k are constants.n(y) is an exponential nonlinear term defined as that is related to the voltage-current relation of the BJT.

Conclusions
The memristor is the fourth fundamental passive circuit element that exhibits a pinched hysteresis loop in the current versus voltage plane.It has attracted much attention of researchers due to applications in several areas of integrated circuit design and computing.Over the last few years, many experiments and mathematical models have been proposed to study the dynamic behaviour of memristive circuits.
In this work, we present some memristive circuits that can exhibit chaotic oscillations.We focus on the simplest, Chua, and Colpitts circuits.The simplest circuit constructed from a capacitor and an inductor connected to a memristor shows periodic and chaotic attractors.By replacing the Chua diode with the memristor, it is possible to observe chaotic attractors and multistability.With regard to the Colpitts circuit, it displays complex nonlinear phenomena by including a memristor.Depending on the parameter, the modified memristor Colpitts circuit exhibits extreme multistability.

Figure 2 :
Figure 2: Temporal evolution of (a) V , (b) w, (c) M , and (d) I.We consider the parameters according to Table1.We observe periodic oscillations.

Figure 3 :
Figure 3: I as a function of V for (a) w0 = 0 nm, (b) w0 = 0.5 nm, (c) w0 = 0.6 nm, and (d) w0 = 0.8 nm.Depending on w0, it is possible to observe the existence of a pinched hysteresis effect, as shown in the panels (b) and (c).

Figure 4 :
Figure 4: Schematic diagram for the three circuit elements in series [12].The circuit is composed of a linear passive inductor (L), a linear passive capacitor (C), and a nonlinear active memristor (M ).

Figure 5
displays y(t) versus x(t) for C = 1.2, L = 3.3, and β = 1.34.The panels (a) and (b) exhibit a periodic attractor for α = 1.15 and a chaotic attractor for α = 0.85, respectively.The chaotic attractor shows sensitivity to initial conditions identified by the spectrum of Lyapunov exponents[23].The system is chaotic when the maximal Lyapunov exponent is positive[24].In Fig.6(a), we present the bifurcation diagram of the maximum values of x(t) as a function of α.The diagram shows period doubling as α increases and a parameter range with chaotic behaviour.By observing the maximal Lyapunov exponent in Fig. 6(b), the negative (periodic attractor) and positive (chaotic attractor) values can be compared with the bifurcation diagram to confirm the periodic and chaotic attractors.
ż = by, ẇ = −cx − dw + x 2 w, where x and y are related to the voltages across the capacitors, z corresponds to the current through the inductor, and w is associated with the internal voltage of the memristor.The memresistance M (w) is defined as α − βw 2 and a, b, c, d, α, and β are the control parameters.Figure 8(a) exhibits three periodic attractors, while Fig. 8(b) displays one periodic attractor (red line) and other chaotic (green line).We consider b = 24, c = 37, d = 12, α = 1.2, β = 0.5, (a) a = 9, and (b) a = 12.In Fig. 9, we compute the bifurcation diagram of the maximum values of z as a function of the control parameter a.The diagram is plotted by forward (black) and backward (red) for b = 24, c = 37, d = 12, α = 1.2, and β = 0.5.It is possible to observe period-doubling cascades to chaos and various narrow windows in which chaos goes to periodic behaviour.We find the existence

Figure 7 :
Figure 7: Schematic diagram for the Chua circuit with a non ideal voltage controlled memristor [15].The memristor replaces the Chua diode.

Figure 9 :
Figure 9: Bifurcation diagram of the local maximum of the dynamical variable z as a function of the control parameter a for b = 24, c = 37, d = 12, α = 1.2, and β = 0.5.The blue and black lines mark the a values used in the phase portraits of Figs.8(a) and 8(b), respectively.

Figure 10 :
Figure 10: Basins of attraction z(0) × x(0) with y(0) = w(0) = 0.0 for attractors depicted in Fig. 8(a) (a = 9) and Fig. 8(b) (a = 12).The white, blue, and red regions correspond to subsets of the initial conditions of different attractors with period one, while the green region corresponds to a chaotic attractor.

Figure 11 :
Figure 11: Schematic diagram for the Colpitts circuit where BJT is a bipolar junction transistor[36].The circuit has a BJT as the gain element and a resonant network composed of a pair of capacitors (C1 and C2) and an inductor (L).