Dynamics Analysis of Memristor Chaotic Circuit with Coexisting Hidden Attractors

In this paper, a novel Sr 0 . 97 Ba 0 . 03 TiO 3 − δ memristor based ﬁfth-order chaotic circuit was proposed. A new cubic nonlinear magnetic control model was established by analyzing the measured data. The equilibrium points and its stability of the circuit system were analyzed by the Jacobi matrix method, and the effects of the initial states and circuit parameters on the system were discussed though meth-ods of Lyapunov exponents spectra, bifurcation diagrams, phase diagrams, and poincar´e maps. The results show that the chaotic circuit can produce complex dynamic phenomena with the variation of initial states and circuit parameters. The dynamic phenomena such as coexisting attractors have been discovered. In particular, coexisting hidden attractors were generated in the chaotic circuit, which was of great signiﬁcance in engineering applications. the unique nonlinear characteristics of memristor,


Introduction
The first physical memristor was reported by the Hewlett-Packard Laboratory in 2008 [1]. It confirmed the existence of the fourth basic circuit element, that is memristor. The memristor have attracted the attention of many scholars for their potential applications in cellular neural network [2,3], associative and non-associative learning [4,5], circuit implemented electronic neurons [6], memory circuit [7], nonlinear oscillating circuits and other fields [8][9][10][11]. Because of the unique nonlinear characteristics of memristor, the chaotic circuit based on memristor has more abundant dynamic behaviors.
Since the chaos phenomenon was discoveried in 1963 [12], many scholars have started to study chaos theory. Chaos circuit can be used as an important way to study chaos theory. Chua proposed the Chuas circuit with a new type of strange attractor [13], which had a very simple structure and generated a variety of dynamic behaviors. Chua and Itoh proposed a new chaotic circuit which uses a memristor to replace the traditional Chua diode in Chuas circuit [14]. Some interesting dynamic phenomena were also observed in the memristor chaotic circuit [15][16][17][18]. For example, multistability is a common phenomenon in many nonlinear dynamical systems [19][20][21][22][23][24][25]. In Ref. [22], a new two-memristor-based chaotic circuit was proposed and the coexistence of multiple attractors was analyzed. The multistability phenomenon of memristor-based chaotic circuits extremely relies on the initial conditions and the circuit parameters, which can produce some behaviors such as coexisting attractors and coexisting limit cycle.
Another kind of attractor newly proposed in recent years was called hidden attractor, whose basin does not intersect with the small neighborhood of any equilibrium points. The coexisting attractors and hidden attractors are of considerable importance in nonlinear dynamics and engineering applications [26][27][28][29][30][31][32]. In Ref. [30], a new three-dimensional quadratic two-wing hidden chaotic system is proposed, and fractional order multi-wing chaotic hidden attractors are generated in the system. In Ref. [31], the system has coexisting hidden attractors for some range of parameters.
In order to better apply the characteristics of physical memristor in chaotic circuits, this paper presents a fifth-order chaotic circuit based on the improved Sr 0.97 Ba 0.03 TiO 3−δ (SBT) memristor which was prepared in our laboratory [33], and the dynamic behaviors of the circuit system were ana-lyzed through various methods. The rest of this paper is organized as follows. In Sect. 2, a modeling of memristor is established. In Sect. 3, The memristor chaotic circuit is designed. In Sect. 4, the stability of the system is analyzed, and the dynamics of dependence on the initial states are studied by means of numerical simulations. In particular, the coexisting hidden attractors can be found in this system. In addition, the influences of circuit parameters on dynamic behaviors of the system are studied in Sect. 5. The conclusions are given in Sect. 6.

Modeling of memristor
The Sr 0.97 Ba 0.03 TiO 3−δ (SBT) nanoscale memristor was fabricated using magnetron sputtering technology [33]. The I-V characteristic of the SBT memristor was tested and the measured data was obtained as shown in Fig. 1, which exhibits an obvious "8" hysteresis curve. It indicates that the SBT memristor can fit the basic requirements of the memristor characteristics well. The measured current I and voltage U were processed using the subsection-average method [34], the average current and average voltage after pre-processing are described in red line. The charge q and flux ϕ was obtained by integrating the current and voltage data respectively, and the resulting discrete point plots of q and ϕ are shown in Fig. 2. The mathematical models for the memristor are commonly the cubic nonlinear model, the quadratic nonlinear model and the segmented linear model. In this paper, the cubic nonlinear model is used to fit the discrete point plots of q and ϕ. The fitting results are shown in Fig. 2(blue line). Among them, the adopted cubic nonlinear model is where a > 0, b > 0.
Therefore, the memristive value W (ϕ) of the memristor is: The fitting result was selected is:  The value of the memristive conductance is as follows: This mathematical model for the SBT memristor was derived as: W (ϕ (t)) = P + Qϕ 2 = 0.018 + 3 * 0.026ϕ 2 (8) where, P = 0.018S, Q = 3 * 0.026S/W b 2 , i (t) is the current flowing through the memristor, u (t) is the voltage across the memristor, ϕ (t) is the magnetic flux of the memristor, and W is the memristive conductance value of the memristor.
Dynamics analysis of memristor chaotic circuit with coexisting hidden attractors 3

Design of memristor chaotic circuit
The circuit diagram of the SBT memristor chaotic circuit is depicted in Fig. 3. The circuit in the Fig. 3 consists of four linear capacitors C 1 , C 2 , C 3 and C 4 , an operational amplifier A, six linear resistors R 1 , R 2 , R 3 , R 4 , R a and R b , a linear conductance G and a nonlinear SBT memristor W (ϕ), in which C 1 , C 2 , C 3 , R 1 , R 2 , R 3 constitute the double T network. In this circuit, an oscillating circuit is composed of the double T network and operational amplifier A, which generate the voltage oscillating signals. When the parameters are R 1 =R 2 =R, R 3 =0.5R, C 1 =C 2 =C, C 3 =2C and the magnification is A = 1 + R a /R b , the oscillation condition is satisfied and chaotic behavior may be generated. According to Ohm's law and Kirchhoff's law, the state equation of the circuit can be obtained as follows: can be expressed to dimensionless dynamical system as follows: Set the variable parameters as: A double-scroll chaotic attractor is generated by MATLAB simulation under the initial condition (0.001, 0.001, 0.015, 0.001, 0.01), as shown in Fig. 4. The five Lyapunov exponents are LE 1 = 0.05875, LE 2 = 0.0.005495, LE 3 = -0.001812, LE 4 = -0.8574, and LE 5 = -11.67, which are calculated by using the Jacobi matrix method. The sum of Lyapunov exponents is negative, which proves that the circuit is chaotic.
4 Dynamic analysis of the memristor circuit

Basic dynamic analysis of the system
The right side of Eq. 10 is equal to 0, then the equilibrium point of the system can be obtained as: where c is an arbitrary constant. All the points on the w-axis can be an equilibrium point. The Jacobian matrix evaluated at the equilibrium point is given as: where The characteristic equation is consequently written as: The parameter w(c) = 0.018+3 * 0.026c 2 , α = 21.5, β = 0.667, γ = 1, η = 1, A = 1, G 0 = −0.6 in Eq. 13: (14) According to the Rouse stability criterion, if the roots of the Eq. 14 satisfy that all the real parts of the real and complex roots are negative, the necessary and sufficient conditions are: By calculation, when |c| > 1.527, the system is in stable state, while the |c| < 1.527, the system equilibrium point is in an unstable state. And the system may exhibit chaos, limit cycle and other nonlinear dynamics behaviors near these unstable equilibrium points.
4.2 Effects of initial states on the dynamic characteristics of the system 4.2.1 Dynamic analysis of dependence on the initial state w(0) When α =21.5, β =0.667, γ =1, η =1, A=1, G 0 =-0.6, the initial state is (0.001, 0.001,0.015, 0.001, w(0)). The variation range of w(0) is from -5 to 5. The Lyapunov exponents spectrum and bifurcation diagram of state variable w is shown in Fig. 5. In order to see the changes of Lyapunov exponents more clearly, the four Lyapunov exponent are shown. The Lyapunov exponents spectrum of the system is highly consistent with the bifurcation diagram.With the increase of the initial state w(0), the system presents a variety of dynamic behaviors. In the interval [-5 Fig. 6. Fig. 6(a) is the limit cycle, Fig. 6(b) is the double scroll chaotic attractors, Fig. 6(c) is the single scroll chaotic attractor, Fig. 6(d) is the sinks. system has the coexistence of single scroll chaotic attractor and limit cycles, the phase diagram and Poincaré section are shown in Fig. 7. When the initial state is (0.001, 0.001, 0.015, 0.001, ±0.2), the system has the coexistence of double scroll chaotic attractors, its phase diagram and Poincaré section are shown in Fig. 8. The Lyapunov exponents spectrum and bifurcation diagram of state variable y is shown in Fig. 9. The Lyapunov exponents spectrum of the system is highly consistent with the bifurcation diagram. When the system parameters remain unchanged and the initial state is (0.001, ±0.298, 0.015, 0.001, 0.01), the coexistence of limit cycles exists in the system. The coexistence phase diagram and Poincaré section diagram are shown in Fig. 10. When the initial state is (0.001, ±0.167, 0.015, 0.001, 0.01), the coexistence of single scroll chaotic attractors exists in the system. The coexistence phase diagram and Poincaré section diagram are shown in Fig. 11. When the initial state is (0.001, ±0.142, 0.015, 0.001, 0.01), the coexistence of the single scroll chaotic attractor and the double scroll chaotic attractor exists in the system. The coexistence phase diagram and the Poincaré section diagram are shown in Fig. 12. When the initial state is (0.001, ±0.036, 0.015, 0.001, 0.01), there is a phenomenon of coexistence of double chaotic attractors in the system. The coexistence phase diagram and the Poincaré section diagram are shown in Fig. 13.  When the initial state x(0), z(0) and u(0) values are changed, the chaotic system will produce the coexistence phenomenon. The initial state parameters and the coexistence behavior are shown in Table 1, and the phase diagram and the Poincaré section diagram of the coexistence phenomenon are shown in Figs. 17, 18, 19, 20, 21 and 22.   The hidden attractors are difficult to find because both basins of attraction and the dimension of the attractor could be very small. Coexisting hidden attractors are coexistence of multiple hidden attractors, which was called hidden multistability. At present, there are few studies on the existence of coexisting hidden attractors in actual chaotic circuits. When   1 shows that the system is chaotic (Fig. 5), so the coexisting hidden attractors in blue line may exist in the system. For example, when w(0) = 2.5, the double scroll chaotic attractor is obtained (Fig. 25). The finite-time local Lyapunov exponents on the time interval t ∈ [0, 50] with w(0)=2.5 are shown in Fig. 23, and they are calculated as LE 1 =0.0128, LE 2 =0.0199, LE 3 = -0.0176, LE 4 = -1.0213, LE 5 = -12.6051, which indicates that the memristor circuit system is chaotic. When w(0) = -2.5, the single scroll chaotic attractor in red line is obtained (Fig. 25). The finitetime local Lyapunov exponents on the time interval t∈[0, 50] with w(0) = -2.5 are shown in Fig. 24, and they are calculated as LE 1 =0.0152, LE 2 =0.0337, LE 3 = -0.0203, LE 4 = -1.0099, LE 5 = -11.9117, which indicates that the memristor circuit system is chaotic. Consequently, the coexisting hidden chaotic attractors exist in the system.

Conclusion
In the paper, the cubic memristor model of SBT memristor is established, and the single-memristor fifth-order chaotic circuit is designed. The equilibrium stability of the circuit is analyzed by numerical calculation method, and the dynamic behaviors of the system with the initial states and circuit parameters is studied by using phase diagram, Lyapunov exponents spectra, bifurcation diagrams and Poincaré maps.
The results show that the system exhibits complex dynamic behaviors such as stable points, chaotic attractors, limit cycles and a variety of coexistence phenomena. Particularly, the system has a special dynamic behavior of coexisting hidden attractors. All the results will contribute to the practical application of chaotic circuits.