Memcapacitor model and its application in chaotic oscillator with

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Memcapacitor model and its application in chaotic oscillator with memristor Guangyi Wang, Shouchi Zang, Xiaoyuan Wang, Fang Yuan, and Herbert Ho-Ching Iu

Citation: Chaos 27, 013110 (2017); doi: 10.1063/1.4973238 View online: http://dx.doi.org/10.1063/1.4973238 View Table of Contents: http://aip.scitation.org/toc/cha/27/1 Published by the American Institute of Physics

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CHAOS 27, 013110 (2017)

Memcapacitor model and its application in chaotic oscillator with memristor Guangyi Wang,1 Shouchi Zang,1 Xiaoyuan Wang,1,a) Fang Yuan,1 and Herbert Ho-Ching Iu2 1

Institute of Modern Circuits and Intelligent Information, Hangzhou Dianzi University, Hangzhou 310018, China 2 School of Electrical, Electronic, and Computer Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

(Received 22 August 2016; accepted 12 December 2016; published online 12 January 2017) Memristors and memcapacitors are two new nonlinear elements with memory. In this paper, we present a Hewlett-Packard memristor model and a charge-controlled memcapacitor model and design a new chaotic oscillator based on the two models for exploring the characteristics of memristors and memcapacitors in nonlinear circuits. Furthermore, many basic dynamical behaviors of the oscillator, including equilibrium sets, Lyapunov exponent spectrums, and bifurcations with various circuit parameters, are investigated theoretically and numerically. Our analysis results show that the proposed oscillator possesses complex dynamics such as an infinite number of equilibria, coexistence oscillation, and multi-stability. Finally, a discrete model of the chaotic oscillator is given and the main statistical properties of this oscillator are verified via Digital Signal Processing chip experiments and National Institute of Standards and Technology tests. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4973238] A memristor is the fourth basic circuit element and memcapacitor is generalized from the memristor. In order to investigate the dynamic behaviors of memristors and memcapacitors in application circuits, this paper presents a strict Hewlett-Packard (HP) memristor model and a charge-controlled memcapacitor model. A chaotic oscillator based on a memristor and memcapacitor is also designed. The dynamic characteristics of the presented chaotic system are analyzed, including equilibrium sets, Lyapunov exponent spectrums, and bifurcations. Coexisting attractors are also found in this system and the basins of attraction are provided. Besides, the presented system is realized by Digital Signal Processing (DSP) chip experiments and National Institute of Standards and Technology (NIST) test of the chaotic system is given.

I. INTRODUCTION

In 1971, Chua predicted the existence of memristors in theory, and the element characteristics, synthetic principles, and applications were elaborated in 1976.1 In 2008, HP (Hewlett-Packard) labs proved the existence of memristors using TiO2 materials and nanotechnology.2 Since then, memristors have attracted widespread attention and become a hot research area. Among memristor based chaotic circuit researches, memristor models were used mostly to help constructing a chaotic circuit. In Refs. 3–6, the memristor models with the smooth quadratic, cubic, and piecewise nonlinear characteristics were used, respectively, to replace the nonlinear resistor in Chua’s circuit to construct a variety of memristor based chaotic circuits. In Ref. 7, the author realized a a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

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memristor based on off-the-shelf components and constructed a chaotic circuit on a breadboard based on it. A Twin-T notch filter feedback controller was designed and employed to control a memristor based chaotic circuit in Ref. 8, and a chaotic circuit based on two realistic nonlinear models of HP Memristor was designed in Ref. 9. On the other hand, memristors are used to construct neural networks. For instance, Ref. 10 presented a general class of memristive neural networks with time delays; a general class of memristive neural networks with discrete and distributed delays is investigated; in Refs. 11 and 12 investigated the exponential lag synchronization control of memristor based neural networks via the fuzzy method and applications in pseudo random sequence generators. In 2009, Ventra and Chua extended the concept of memristors to memcapacitors and meminductors.13 Similar to the fact that memristors can store information without power supply, these two elements can be applied as non-volatile memory elements and adapt to learning and spontaneous behavior simulation. Compared with the researches on memristor based oscillators, there are very few studies on memcapacitor based oscillators in recent years. In Ref. 13, the concept of memcapacitors was introduced and a piecewise linear model was proposed. In Ref. 14 a memcapacitor circuit model based on a LDR (Light Dependent Resistor) memristor was proposed. Also, different behavioral models of memcapacitors were developed and implemented in SPICE-compatible simulators in Refs. 15–17, and mutators were used to transform the memristor into the memcapacitor in Ref. 18. On the other hand, Ref. 19 introduced a new memcapacitor emulator without using any memristor, and a floating memcapacitor emulator without grounded restriction was designed, which can be practically applied in electronic circuits.20 Moreover, for exploring the properties of the memristor, memcapacitor, and meminductor in nonlinear circuits, some oscillators

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based on the three memory elements were presented,21 and some special dynamical characteristics such as multi attractors,22 coexistence attractors,23,24 hidden attractors,23,25 and extreme multistability26 were found. In addition, these special properties were also found in other chaotic systems.27–33 In the present studies, most of the chaotic circuits only adopt one memory element, and very few chaotic oscillator circuits use more than two different memory elements. In this paper, a new chaotic oscillator with a HP memristor and a memcapacitor is proposed for exploring the properties of the memristor and memcapacitor in nonlinear circuits. The equilibrium point sets, bifurcation diagrams, and Lyapunov exponent were calculated to analyze the dynamic behavior of the circuit. Moreover, the DSP technology is used to realize this chaotic oscillation circuit, and the experimental results verified the correctness of the theoretical analysis, which has certain referential significance for the practical applications of the memristor and memcapacitor. II. TiO2 MEMRISTOR AND MEMCAPACITOR

w ðt Þ i ðt Þ RON dwðtÞ RON vON D ¼ lv iðtÞ ¼ lv ¼ lv EON ; ¼ lv w ðt Þ D wðtÞ dt (5) where EON ¼ vON =wðtÞ is the field intensity in the doped region of the memristor and vON ¼ iðtÞRON wðtÞ=D is the voltage across the region; RON wðtÞ=D is the resistance of the doped region. Considering iðtÞ ¼ vðtÞ=MðqÞ, From Eq. (5), the voltage across the doped region can be depicted as w ðt Þ D vðtÞ: vON ðtÞ ¼ ROFF  RON wðtÞ ROFF  D RON

The HP TiO2 memristor model shown in Fig. 1 is currently the most widely used model. It is composed of two parts, i.e., the undoped part whose resistance is ROFF and the doped part whose resistance is RON. D is the length of the memristor, w(t) is the width of the doped region, and lv denotes the dopant mobility. The mathematical model of the HP TiO2 memristor is   wðtÞ w ðt Þ þ ROFF 1  ; (1) MðtÞ ¼ RON D D (2)

(6)

Inserting Eq. (6) into Eq. (5), we can get the following equation: dwðtÞ ¼ dt

A. Flux-controlled model of the HP memristor

dwðtÞ RON iðtÞ: ¼ lv D dt

Therefore, we can rewrite Eq. (2) as follows:

RON vðtÞ D : ROFF  RON w ðt Þ ROFF  D lv

(7)

Integrating Eq. (7) with respect to time t, the relation between wðtÞ and uðtÞ can be obtained as 1 2 ðRON  ROFF ÞwðtÞ þ DROFF wðtÞ  lv RON uðtÞ ¼ 0; (8) 2 Ðt where uðtÞ ¼ t vðsÞds. By solving Eq. (8), we can get 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DROFF 6 D2 R2OFF þ 2ðRON  ROFF Þlv RON uðtÞ : wðtÞ ¼ RON  ROFF (9)

(3)

By inserting Eq. (9) into Eq. (4), the memristance of the flux-controlled memristor is obtained as below pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (10) MðuÞ ¼ 6 k1 þ k2 u;

where q(t) is the integral of the current to time and for simplicity, in this paper w0 is set to zero. By inserting Eq. (3) into Eq. (1), the memristance of the memristor can be obtained as

where k1 ¼ R2OFF and k2 ¼ 2ðRON  ROFF Þlv RON =D2 . In Eq. (10), the positive sign denotes passive memristor and the negative sign represents the active memristor that may not be considered here. From Eq. (10), we can get the memristance of the flux-controlled TiO2 memristor as follows:

The integral of Eq. (2) is written as the following: wðtÞ ¼ lv

RON qðtÞ þ w0 ; D

MðqÞ ¼ ROFF 

ROFF  RON wðtÞ: D

(4)

1 W ðuÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : k1 þ k2 u

(11)

In order to test the characteristics of the flux-controlled memristor, let RON ¼ 10 X, ROFF/RON ¼ 100 X, lv ¼ 1010 cm2 s1 v1, D ¼ 10 nm, k1 ¼ 1 k X2, and k2 ¼ 1.98 k X s1 v1, the voltage across the memristor is shown in Fig. 2(a), whose amplitude is 1 V and frequency is 1 Hz, and the v - i characteristic which is a pinched hysteresis loop is shown in Fig. 2(b). B. Charge-controlled model of a new memcapacitor

Eq. (1) can be rewritten as

FIG. 1. Structure of the HP TiO2 memristor.

MðtÞ ¼ ROFF  ðROFF  RON Þ

wðtÞ ¼ a  bqðtÞ: D

(12)

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FIG. 2. Characteristics of the fluxcontrolled memristor: (a) voltage and current on the memristor and (b) voltage-current hysteretic curves of the memristor.

According to the relationship, vðtÞ ¼ MðqÞiðtÞ, and considering Eq. (12), we obtain ! ðt (13) vðtÞ ¼ ða  bqðtÞÞiðtÞ ¼ a  b iðsÞds iðtÞ; t0

where a ¼ ROFF and b ¼ ðROFF  RON Þlv RON =D2 . According to Eq. (13), we can write an expression of memductance as follows: ! ðt (14) iðtÞ ¼ c þ d vðsÞds vðtÞ; t0

where c and d are constants related to the parameters of the memristor. Memristor is a passive two-terminal device whose property depends on the past state of the system. Reference 13 defined the memcapacitor as a charge-controlled capacitor, and the relation between voltage vc and charge q was described as ! ðt 1 qðsÞds qðtÞ; (15) vc ðtÞ ¼ CM t0

CM1

is the inverse memcapacitance. where Considering Eqs. (14) and (15), we can define a chargecontrolled memcapacitor ! ðt (16) vc ðtÞ ¼ a þ b qðsÞds qðtÞ ¼ ða þ brÞqðtÞ: t0

Inverse of the memcapacitance can be described as CM 1 ðrÞ ¼ a þ br; where r ¼ time.

Ðt t0

(17)

qðsÞds is the integral of charge with respect to

In order to test the feature of the charge-controlled memcapacitor, we set a ¼ 1.5 F1, b ¼ 1.4 F1 C1 s1, and q(t) ¼ Qmsin(2pft) C, and the simulated voltage across the memcapacitor is shown in Fig. 3(a), where the amplitude of the charge is 1 C and its frequency is 1 Hz. The q-v hysteresis curve is shown in Fig. 3(b). III. THE CHAOTIC CIRCUIT CONTAINING A HP MEMRISTOR AND A MEMCAPACITOR

Based on the models of the HP memristor in Eq. (11) and the memcapacitor in Eq. (16), a new chaotic circuit is designed as shown in Fig. 4. By applying Kirchhoff’s laws to the circuit in Fig. 4, the state equations for the voltage vC across the linear capacitor C, the current iL through the linear inductor L and the charge qcm on the nonlinear memcapacitor Cm are described by 8 dvC > > ¼ i L  W ðu Þv C >C > > dt > < diL (18) ¼ vcm  vC L > dt > > > > dq > : cm ¼ Gvcm  iL : dt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In terms of Eqs. (11) and (16), i.e., WðuÞ ¼ 1= k1 þ k2 u and vcm ¼ ða þ brcm Þqcm , Eq. (18) becomes 8  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dvC > > ¼ i  1= k1 þ k2 u vC C > L > > dt > < diL (19) ¼ ða þ brcm Þqcm  vC L > dt > > > > dq > : cm ¼ Gða þ brcm Þqcm  iL ; dt

FIG. 3. Characteristics of the chargecontrolled memcapacitor: (a) waveforms of voltage and charge and (b) voltage-charge hysteretic curve of the memcapacitor.

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IV. EQUILIBRIUM POINTS

FIG. 4. Oscillation circuit based on TiO2 memristor and memcapacitor models.

where the flux u and the variable rcm are the intermediate variables for the memristor M and the memcapacitor Cm, respectively. In order to be able to solve the set of three differential equations, two equations with respect to u and rcm must be added into Eq. (19), because it has three equations but with five variables. Therefore, Eq. (19) can be transformed into the form 8  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > dv > > C C ¼ iL  1= k1 þ k2 u vC > > dt > > > > diL > > ¼ ða þ brcm Þqcm  vC L > > > dt > < dqcm (20) ¼ Gða þ brcm Þqcm  iL > dt > > > > du > > ¼ vC > > dt > > > > drcm > > ¼ qcm : : dt If we set 1/C ¼ 2.5 F1, 1/L ¼ 1/1.8 H1, G ¼ 1 mS, k1 ¼ 1 kX2, k2 ¼ 1.98 kX s1 v1, a ¼ 1.5 F1 and b ¼ 1.4 F1 C1 s1, and give the initial condition (vC ð0Þ; iL ð0Þ; qcm ð0Þ; uð0Þ; rcm ð0Þ) as (0.1, 0.1, 0.1, 0.04, 0.04), the chaotic attractor can be obtained as shown in Figs. 5(a) and 5(b) by solving Eq. (20). Notice that in Fig. 4 we can get the relations as vM ¼ vC, iM ¼ iL–iC, so the chaotic pinched hysteresis curves of the flux-controlled memristor for describing the relationship vM–iM can be obtained as shown in Fig. 5(c). Similarly, Fig. 5(d) exhibits the chaotic pinched hysteresis curves of the charge-controlled memcapacitor for the relation between qcm and vcm where vcm ¼ ða þ brcm Þqcm . Fig. 6(a) shows the Poincare section when qcm equals 0 (projected on vC–iL plane), from which we can learn that the motion of the system is a continuous curve in the Poincare mapping. The time domain waveforms of vC, iL, and qcm are described in Fig. 6(b), which are aperiodic and pseudorandom. All the above results show that the system we built using the TiO2 memristor and the proposed memcapacitor becomes a chaotic oscillator. Lyapunov exponents describe motion trajectory mutually exclusive and mutually attractive features. The maximum Lyapunov is important for judging a chaotic system. Using the Jacobian method, from Eq. (18) we can obtain the Lyapunov exponent set which is (0.1668, 0.0040, 0.00089224, 0.0089, 2.9233) and shows that the system is chaotic.

Let v_ C ¼ i_L ¼ q_ cm ¼ u_ ¼ r_ cm ¼ 0 in Eq. (18). The equilibrium points of Eq. (18) are s ¼ {(vC, iL, qcm, u, rcm)j vC ¼ iL ¼ qcm ¼ 0, u > k1/k2, rcm ¼ R ¼ constant}, which is an equilibrium set and has an infinite number of equilibrium points. The Jacobian matrix J at this equilibrium set is given as 2 3 1 W 0 0 0 6 7 C 6 7 6 1 7 6 7 6 L 0 A 0 07 J¼6 (21) 7; 6 0 1 B 0 0 7 6 7 6 7 0 0 0 05 4 1 0 0 1 0 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where W ¼ 1=C k1 þ k2 u, A ¼ 1=Lða þ bRÞ, and B ¼ G ða þ bRÞ, and its characteristic equation is given as k2 ðk3 þ a1 k2 þ a2 k þ a3 Þ ¼ 0;

(22)

where a1 ¼ ðW þ BÞ, a2 ¼ A þ BW þ 1=LC, and a3 ¼ ðAW þ B=LCÞ. When Eq. (22) has at least one root of the real part which is greater than zero, the equilibrium point is an unstable equilibrium point. Obviously, the coefficients of the three polynomials in Eq. (22) are non zero. Based on the RouthHurwitz criterion, the system is stable on the conditions as follows:    a1 1 0    Dk ¼  a3 a2 a1  > 0; (23)  0 0 a3  where k ¼ 1, 2, and 3, that is 8 > < D1 ¼ a1 > 0 D2 ¼ a1 a2  a3 > 0 > : D3 ¼ a3 ða1 a2  a3 Þ > 0:

(24)

In general, if the system is in chaos or hyperchaos, there should be at least one positive eigenvalue (or the real part of the eigenvalue), which leads to the consequence that the expressions of Eq. (24) are not completely positive. For instance, if we set 1/C ¼ 2.5 F1, 1/L ¼ 1.8 H1, G ¼ 1 mS, k1 ¼ 1 kX2, k2 ¼ 1.98 kX s1 v1, a ¼ 1.5 F1, b ¼ 1.4 F1 C1 s1, u ¼ 0, and R ¼ 0.4 kX, then D1 ¼ 0.44, D2 ¼ 1.18356, and D3 ¼ 2.534. In this condition, the equilibrium points are unstable and possess the characteristic of a chaotic system. V. INFLUENCE OF PARAMETERS TO THE SYSTEM DYNAMICS A. Influence of parameter 1/C

The Lyapunov exponent spectrum versus parameter 1/C and the corresponding bifurcation diagram are shown in

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FIG. 5. Chaotic attractors of the oscillator and chaotic pinched loops of the memristor and the memcapacitor: (a) projection of the attractor on the vC-iL plane, (b) projection of the attractor on the vC-qcm plane, (c) chaotic pinched hysteresis curves of the HP memristor on the vM-iM plane, and (d) chaotic pinched hysteresis curves of the memcapacitor on the qcmvcm plane.

FIG. 6. (a) Poincare section when qcm equals 0 and (b) the time domain waveforms of vC, iL, and qcm.

Fig. 7 (for clarity, the LE5 is not drawn, because of the LE5 is too small, similarly hereinafter) and Fig. 8, respectively, where 1/L ¼ 1.8 H1, a ¼ 1.5 F1, b ¼ 1.4 F1 C1 s1, G ¼ 1 mS, k1 ¼ 1 kX2, k2 ¼ 1.98 kX s1 v1, 1/C2[2, 2.5] and the initial condition (vC ð0Þ; iL ð0Þ; qcm ð0Þ; uð0Þ; rcm ð0Þ) is (0.1, 0.1, 0.1, 0.04, 0.04). From Figs. 7 and 8, it can be seen that the system gradually transits from chaotic to periodic. When 1/C is equal to 2.7 F1, 2.55 F1, and 3.1 F1, the system is in the state of chaos shown in Figs. 9(a), 9(c), and 9(e). When 1/C is equal to 2.37 F1 and 2.766 F1, the system is in period doubling trajectory shown in Figs. 9(b) and 9(d), respectively. When 1/C is equal to 5 F1, the system is in a periodic trajectory shown in Fig. 9(f). In particular, the system displays a weak hyperchaotic state with two positive Lyapunov exponents when 1/C2[2.0, 2.8].

qcm ð0Þ; uð0Þ; rcm ð0Þ) as (0.1, 0.1, 0.1, 0.04, 0.04), we can obtain the Lyapunov exponent spectrum with respect to the parameter 1/L shown in Fig. 10 and the bifurcation diagram in Fig. 11. With the increase of the parameter 1/L, the system is changed to chaotic trajectory by a short period bifurcation.

B. Influence of parameter 1/L

By setting 1/C ¼ 2.5 F1, a ¼ 1.5 F1, b ¼ 1.4 F1 C1 s , G ¼ 1 mS, k1 ¼ 1 kX2, k2 ¼ 1.98 kX s1 v1, and 1/L2 [1.68, 2.2] H1, under the initial conditions (vC ð0Þ; iL ð0Þ; 1

FIG. 7. Lyapunov exponents versus parameter 1/C.

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FIG. 8. Bifurcation of vC versus parameter 1/C.

FIG. 10. Lyapunov exponents versus parameter 1/L.

Then, the system transforms to a periodic trajectory. In the region [1.68, 2.2], the system goes through several cycle windows such as [1.756, 1.781], [1.851, 1.866], and [1.945, 1.958] and eventually evolves into a periodic trajectory

by an inverse period-doubling bifurcation. The attractor evolutions of the system with the parameter 1/L are shown in Fig. 12, where the values of 1/L are 1.7 H1, 2.06 H1, 2.1 H1, and 2.2 H1.

FIG. 9. The phase diagram of the system with the change of the parameter 1/C: (a) 1/C ¼ 2.2, (b) 1/C ¼ 2.37, (c) 1/C ¼ 2.55, (d) 1/C ¼ 2.766, (e) 1/C ¼ 3.1, and (f) 1/C ¼ 5.

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FIG. 11. Bifurcation of vC versus parameter 1/L. FIG. 13. Dynamical map with parameters a and b.

C. Influence of memcapacitor parameters a and b

The dynamical characteristics of the system will vary with the changes of the internal parameters a and b of the memcapacitor. The dynamical map of the system versus a and b is given in Fig. 13, which depicts the dependence of the dynamic behavior on the memcapacitor parameters. In Fig. 13, the yellow areas indicate the periodic state, the blue regions represent the chaotic state, and the red areas show unbounded zones. In blue regions, some of the deep blue regions indicate the hyperchaotic state. The relatively concentrated and complete areas in the dynamical map such as blue chaotic region or yellow periodic region show that the system has a robust solution set, namely, when the parameters a and b change within certain range, the system can run in a continuous and stable state such as hyperchaos, chaos, and period. From Fig. 13, we can see that when the parameters a and b increase, the system evolves from the

periodic region to the chaotic or hyperchaotic region. The finely banded yellow areas in the dynamical map indicate that the system bifurcated from periodic to chaotic. D. Effects of initial conditions on dynamical characteristics

Unlike some general memristor based oscillators which have the same oscillation state such as chaos or period when they have different initial values but with the same system parameters, the proposed system shows a special feature, i.e., it can generate different oscillation states with the same circuit parameters but under different initial values of the state variables. We call the phenomenon as co-oscillation. Let 1/C ¼ 2.5 F1, 1/L ¼ 1.8 H1, G ¼ 1 mS, k1 ¼ 1 kX2, k2 ¼ 1.98 kX s1 v1, a ¼ 1.5 F1, b ¼ 1.4 F1 C1 s1, and vC(0) changes in the initial conditions (vC(0), 0.1,

FIG. 12. The attractor evolutions of the system with the change of the parameter 1/L: (a) 1/L ¼ 1.7, (b) 1/L ¼ 2.06, (c) 1/L ¼ 2.1, and (d) 1/L ¼ 2.2.

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FIG. 14. Lyapunov exponents versus parameter vc(0).

0.1,0.04, 0.04). When vC(0) 2 [0, 0.18], the Lyapunov exponent spectra and the corresponding bifurcation versus vC(0) are shown in Figs. 14 and 15, respectively, which show the relationship between the variable iL and the initial value vC(0). As shown in Figs. 14 and 15, the state of the system evolves from the periodic oscillation to the chaotic oscillation with the increase of vC(0). This shows that the proposed chaotic oscillator containing a memristor and a memcapacitor depends on not only the system parameters but also the initial conditions, thereby appearing a state of co-oscillation or co-attractor. Fig. 16 is an attracting basin, which indicates the different attractors of the system with the same system parameters but different initial conditions. Let 1/C ¼ 2.5 F1, 1/L ¼ 1.8 H1, G ¼ 1 mS, k1 ¼ 1 kX2, k2 ¼ 1.98 kX s1 v1, a ¼ 1.5 F1, b ¼ 1.4 F1 C1 s1, iL(0) ¼ 0.1 mA, u(0) ¼ 0.04 Wb, and rcm(0) ¼ 0.04 Cs, and selecting the initial value of vC(0) and qcm(0) in different regions, the system will generate chaotic and periodic attractors. The projections of the corresponding attractors are shown in Fig. 17. Specifically, Fig. 17(a) corresponds to region (A) in Fig. 16 where the initial condition is (0.06, 0.1, 0.08, 0.04, 0.04); Fig. 17(b) corresponds to region (B) in Fig. 16 where the initial condition is (0.06, 0.1, 0.09, 0.04, 0.04); Fig. 17(c) corresponds to region (C) in

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FIG. 16. The basin of attraction for the chaotic system when iL equals 0.1 mA, u equals 0.04 Wb, and rcm equals 0.04 Cs.

Fig. 16 where the initial condition is (0.1, 0.1, 0.095, 0.04, 0.04); and Fig. 17(d) corresponds to region (D) in Fig. 16 where the initial condition is (0.1, 0.1, 0.107, 0.04, 0.04). The basin of attraction mentioned above is an important tool for the analysis of coexisting attractors, and the multiple steady states depend on the different basins of attraction.24,26 Under different conditions, the shapes of the attractors are different. The coexistences of chaotic attractors and limit cycle are the basic properties of the oscillator. In addition, the oscillator also exhibits other coexisting attractors, such as the coexistence point attractor. For more details, in some initial value combinations, chaotic attractors can coexist with point attractors as shown in Fig. 18(a). While on the conditions of other initial values, a chaotic attractor can coexist with the other chaotic attractor and a limit cycle as shown in Figs. 18(b) and 18(c). Furthermore, Fig. 18(d) shows the coexistence of two different limit cycles, and Fig. 18(e) describes the coexistence of a limit cycle and a point attractor in different initial values. The typical coexisting attractors shown in Fig. 18 and the corresponding initial conditions are described in Table I. VI. DSP IMPLEMENTATION OF THE CHAOTIC SYSTEM

DSP is a digital processor for processing digital systems or discrete system and it can achieve D/A conversion quickly. In order to realize the chaotic oscillation system with a HP memristor and a memcapacitor, we need to convert the continuous chaotic system into a discrete chaotic system. In this paper, a simple Euler algorithm is used to achieve the discretization of the chaotic system. The Euler algorithm is mainly based on the definition of the derivative f 0 ðxÞ ¼ lim

Dt!0

xðtn þ DtÞ–xðtn Þ xnþ1 –xn ¼ lim : Dt!0 Dt Dt

(25)

When Dt tends to zero, Eq. (25) can be approximated as

FIG. 15. Bifurcation of iL versus parameter vc(0).

f 0 ð xÞ 

xnþ1  xn xðn þ 1Þ  xðnÞ ¼ : Dt Dt

(26)

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FIG. 17. Attractors of the system under different initial conditions: (a) (0.06, 0.1, 0.08, 0.04, 0.04), (b) (0.06, 0.1, 0.09, 0.04, 0.04), (c) (0.1, 0.1, 0.095, 0.04, 0.04), and (d) (0.1, 0.1, 0.107, 0.04, 0.04).

FIG. 18. Coexisting attractors on the vC-qcm plane: (a) coexisting chaotic attractor and point attractor, (b) two kinds of coexisting chaotic attractors, (c) coexisting chaotic attractor and limit cycle, (d) two kinds of coexisting limit cycles, and (e) coexisting limit cycle and point attractor.

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TABLE I. Coexisting attractors for various initial conditions. Species of coexisting attractors

Initial conditions

(0.1; 0.1; 0.1; 0.04; 0.04) (0.04; 0.1; 0.1; 0.04; 0.04) Two kinds of chaotic attractors (0.1; 0.1; 0.1; 0.04; 0.04) (0.1; 0.1; 0.1; 0.04; 0.09) Chaotic attractors and limit cycles (0.1; 0.1; 0.1; 0.04; 0.04) (0; 0.1; 0.1; 0.04; 0.04) Two kinds of limit cycles (0.06; 0.1; 0.12; 0.04; 0.09) (0.06; 0.1; 0.08; 0.04; 0.04) Limit cycle and point attractors (0; 0.1; 0.1; 0.04; 0.04) (0.4; 0.1; 0.1; 0.04; 0.04) Chaotic attractors and point attractors

Figures Fig. 18(a) Fig. 18(b) Fig. 18(c) Fig. 18(d) Fig. 18(e)

According to Eq. (26), Eq. (20) can be discretized into the following equations: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > vC ðn þ 1Þ ¼ 1=CðiL ðnÞ  1= k1 þ k2 uðnÞvC ðnÞÞDt þ vC ðnÞ > > > > > > < iL ðn þ 1Þ ¼ 1=Lðða þ brcm ðnÞÞqcm ðnÞ  vc ðnÞÞDt þ iL ðnÞ qcm ðn þ 1Þ ¼ Gða þ brcm ðnÞÞqcm ðnÞDt  iL ðnÞDt þ qcm ðnÞ > > > > uðn þ 1Þ ¼ vC ðnÞDt þ uðnÞ > > > : r ðn þ 1Þ ¼ q ðnÞDt þ q ðnÞ: cm cm cm

(27) Let 1/C ¼ 2.5 F1, 1/L ¼ 1.8 H1, G ¼ 1 mS, k1 ¼ 1 kX2, k2 ¼ 1.98 kX s1 v1, a ¼ 1.5 F1, b ¼ 1.4 F1 C1 s1 and Dt ¼ 0.0001, and the initial condition is vC(0) ¼ 0.1, iL(0) ¼ 0.1, qcm(0) ¼ 0.1, u(0) ¼ 0.04, and rcm(0) ¼ 0.04. After solving the discrete equation in DSP, the obtained digital signal is transformed into analog signal by using the D/A convertor and the output results can be observed by an

oscilloscope, as shown in Fig. 19. Note that Fig. 19(e) is the digital signal before it is converted into the analog signal. VII. NIST TEST OF THE CHAOTIC SYSTEM

In this paper, the proposed chaotic oscillation can be used to generate digital pseudo-random sequences, which can be used in the various information security systems. The randomness of the chaotic sequence has a direct impact on the security of the chaotic encryption algorithm. For the purpose of verifying the randomness of the chaotic system, the test of the binary sequences can be performed by means of the NIST test suite.34 The NIST test suite is the most authoritative tool for pseudorandom test currently, which is a statistical software package and consists of 15 tests. The 2.0 version of test suite package is used in this paper. The test binary sequences are generated by a continuous chaotic real valued signal, which is from the solution sequence vc of Eq. (27). Suppose that the decimal solution sequence is expressed as a1 a2 a3    ai  b1 b2 b3    bj , where a1 a2 a3    ai denotes the integral part and b1 b2 b3    bj denotes the decimal part, a way to generate a binary sequence is by using the following threshold function:  0; if bj < 5 xðbj Þ ¼ (28) 1; if bj  5: For a generated binary sequence, which contains “0” and “1” of a given length n, it is divided into k nonoverlapping parts where each length is m (k ¼ n/m), and n ¼ 1 000 000 000 and m ¼ 1000. This work of the grouping is automatically finished by the NIST test suite package.

FIG. 19. Experimental attractors of the system using DSP technology: (a) vC-iL plane, (b) vC-qcm plane, (c) vM-iM plane, (d) qcm-vcm plane, and (e) chaotic PN sequence digitalized from vC.

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TABLE II. The test report for the proposed oscillator and the Lorenz oscillator. P-value Statistical test Terms Frequency Block frequency Cumulative sums Runs Longest run Rank FFT Non overlapping template Overlapping template Universal Approximate entropy Random excursions Random excursions variant serial Linear complexity

Proportion

Proposed system

Lorenz system

Proposed system

Lorenz system

0.647530 0.940080 0.440975 0.006019 0.224821 0.051281 0.109435 0.670396 0.304126 0.552383 0.587274 0.020315 0.895359 0.005516 0.678686

0.062427 0.021701 0.032489 0.965083 0.296834 0.599693 0.961039 0.568739 0.254411 0.140453 0.912724 0.698671 0.454759 0.554420 0.023228

0.9900 0.9930 0.9900 0.9950 0.9960 0.9870 0.9910 0.9970 0.9860 0.9900 0.9920 0.9950 0.9983 0.9880 0.9930

0.9920 0.9950 0.9910 0.9900 0.9950 0.9820 0.9870 0.9840 0.9910 0.9870 0.9900 0.9813 0.9860 0.9890 0.9900

The sample size for the test sequences is tied to the choice of the significance level a. NIST recommends that the user should fix the significance level a to be at least 0.001 for the NIST tests but no larger than 0.01. A sample size, which is disproportional to the significance level a, may not be suitable. If, for instance, the significance level a is chosen to be 0.001, then it is implied that 1 out of every 1000 sequences will be rejected. If a sample has only 100 sequences, it would be rare to observe a rejection. In this case, the conclusion may be drawn that a sample should be on the order of a-1. That is, for a level of 0.001, a sample should have at least 1000 sequences; while for a level of 0.01, a sample should have at least 100 sequences. For the NIST test, the chaotic sequences extracted from the proposed system are divided into one thousand groups, and each group contains one hundred thousand bits, and we select the significant level a to be 0.01. The analysis report is shown in Table II, where for comparison, the randomness of the well-known Lorenz oscillator is also tested by the NIST suite under the same test conditions and its analysis report is also entered in Table II. From Table II, all the tested P-values of the proposed oscillator satisfy the requirement. If a valuep offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi proportion falls outside the interval defined by 1  a63 að1  aÞ=m, where a is the significance level and m is the sample size, then there is evidence that the data are non-random. In the test, a ¼ 0.01, m ¼ 1000, and the confidence interval is (0.98056, 0.99944). From Table II, we can see that all the tested proportions of the proposed oscillator fall into the confidence interval. The tested results show that random characteristics of the oscillator are definitely up to the standards of the NIST and are better than those of the well-known Lorenz oscillator for most tested terms. Therefore, this oscillator can be used to design pseudorandom sequence generators for various applications of information safety field.

VIII. CONCLUSIONS

This paper proposes a charge-controlled model of HP memristor and a new charge-controlled memcapacitor model. Based on the two models, a chaotic oscillator is designed. Numerical simulation results show that the circuit can generate a wealth of chaotic dynamic behaviors under different parameters. States of the chaotic system are sensitive to the initial conditions of the state variables, hence yielding the special phenomena of coexistence oscillation and its multi-stability. The coexistence oscillation can be used as pseudo sequence generator for generating multi pseudo signals. The results of the DSP experiment and the NIST tests for the proposed system also indicate that this system can exhibit good randomness that is better than the well-known Lorenz system. So the chaotic system has potential application values in the fields of secure communication, chaotic cryptography, and so on. ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China under Grant Nos. 61271064, 60971046, and 61401134 and the Natural Science Foundations of Zhejiang Province, China under Grant Nos. LZ12F01001 and LQ14F010008. 1

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