Chaotic Circuit based on Memristive Elements Chaotic Circuit based on Memristive Elements

Chaotic Circuit based on Memristive Elements Yongbin YuSchool of Information and Software Engineering University of Electronic Science and Technology of ChinaChengdu, CNijing YangSchool of Information and Software EngineeringUniversity of Electronic Science and Technology of ChinaChengdu, ChinaYang_NTashi Nyima School of Information Science and TechnologyTibet UniversityTibet, C Huihui MaSchool of Information and Software Engineering University of Electronic Science and Technology of ChinaChengdu, Chinama_Yancheng WangSchool of Information and Software EngineeringUniversity of Electronic Science and Technology of ChinaChengdu, CMingxiao WangSchool of Electrical and Computer EngineeringThe Ohio State UniversityColumbus, USA Chenyu YangSchool of Information and Software EngineeringUniversity of Electronic Science and Technology of China Chengdu, Chinachenyu_yang_Xiangxiang WangSchool of Information and Software Engineering University of Electronic Science and Technology of ChinaChengdu, CAbstractA novel memristive elements-based chaotic circuit only containing three elements of memristor, memcapacitor and meminductor is highlighted in this paper. The state equations of the memristive circuit are described, and the dynamical behaviors such as equilibrium set, Lyapunov exponents and bifurcation diagram are revealed by theoretical analyses and numerical simulations. Simulation results demonstrate that the sum of five Lyapunov exponents in this circuit is negative and the maximal Lyapunov exponent is equal to 0.1773, which verifies typical characteristics of limit cycle and 1-scroll chaotic attractor. The most striking feature is that this system to build a high-dimensional dynamic equation only using three circuit elements and exhibit rich phenomenon of period and chaotic behavior. Moreover, equivalent realization circuit are design to verify the theoretical analyses and numerical simulations. Keywordsmemristive elements, chaotic circuit, dynamical behavior, experimental simulation, lyapunov exponentsI. Introduction The fourth basic element is proposed by professor Leon Chua in 1971 1. In 1976 2, Chua and Kang generalize the memristor and investigate it to a broader class of nonlinear dynamical systems called memristive systems. It is still a theory until HP labs realize the memristor in 2008 3. After that, in 2009 4, Ventra M.D extends the concept of memristor to memcapacitor and meminductor. Generally, memristor, memcapacitor and meminductor are called memristive elements. In addition, memristor can be widely applied to various fields like RRAM 5, filter circuits 6, neural networks 7-8 and chaotic circuits 9-11. In this paper, the memristive elements including memristor, memcapacitor and meminductor are utilized to design chaotic circuit.The memristive elements are nonlinear element which can easily be used to design chaotic signal generator 2. So many researchers are focused on the memristive chaotic circuits 11-15. In 2008 10, Itoh and Chua propose several nonlinear oscillators based on Chuas circuit by replacing Chuas diode with monotone-increasing and piecewise-linear memristor. In 2010, Muthuswamy and Chua present a simplest chaotic circuit that including three elements of the memristor, inductor and capacitor 11. Chaotic circuits based on memcapacitor and meminductor are presented exclusively 13-18. However, as we known, there are no papers have been published describing chaotic circuit based only on the memristive elements. Chaotic circuit is widely used to various fields encryption and decryption systems 19, 20 and neural networks 21-23. The performance of the encryption and decryption system will be better if the dimension of the chaotic circuit is higher or the behavior is richer 24,25. Moreover, the memristive elements are dynamical elements with nonlinear characteristics. Therefore, there is no need designing extra nonlinear components when build to chaotic circuit. In this paper, a high-dimensional chaotic circuit only using three circuit elements is presented, from which periodic and chaotic behaviors are exhibits.In this paper, by parallel connection circuit of the memristive elements the novel chaotic circuit is designed. In this cases, complex nonlinear dynamic behaviors are investigated by using the Phase Diagrams, equilibrium set, Lyapunov exponent, bifurcation diagram. Furthermore, the electronic circuit is simulated to verify dynamical chaotic behaviors of the memristive elements based chaotic circuit.The rest of this paper is organized in the following manner. The section mainly introduced the prerequisite knowledge. Then, Section mainly designed the new chaotic circuit based on the memristive elements and analyze the equilibrium set of the chaotic circuit, numerical simulation and dynamical behavior. The experimental results of the circuit are displayed in section . Finally, the conclusions are drawn in section .TABLE I. Comparisons Of Memristive Elements-Based Chaotic CircuitItemRef.16Ref.14Ref.18Proposed circuitLinear element1 resistor1 capacitor1 Negative conductance1 resistor1 capacitor1 Negative conductance2 resistor2 capacitorNoMemristive elements1 memristor1 memcapacitor1 memcapacitor1 meminductor1 meminductor1 memristor1 memcapacitor1 meminductorThe number of componentsLimit cycleChaotic attractorsFiveFiveThreeDimensionFiveFiveFourFiveDynamic behaviorLimit cycleChaotic attractorsTwo kinds of chaotic attractorsLimit cycleChaotic attractorsLimit cycleChaotic attractorsLimit cycleChaotic attractorsII. PreliminariesThe models of the memristive elements, memristor 26, memcapacitor 12 and meminductor 14. The mathematical model of memristor is proposed as follows where the W(z) denotes memductance of memristor, a and b are the internal constant of W(z), vM denotes voltage of memristor, z is the internal state of memristor and a is the constant.The memcapacitance which represents the relation of flux and charge is configured as charge controlled as follows The derivative equation form (2) with respect to time t can be obtained as follows where the parameters a and b are constant, A smooth continuous cubic monotone-increasing nonlinearity meminductor is described by The derivative equation form (4) with respect to time t can be obtained where c and d are constant, . The three models are applied into this paper to build a new chaotic circuit.III. The Chaotic Circuit Memristive ElementThe comparisons of some chaotic circuit of memristive elements are described in Table 1. It is obvious from the table that the chaotic circuit proposed in this paper has a simple circuit structure only using three memristive elements to build a high-dimensional equation. This system has rich dynamic behavior. In the next work, the chaotic circuit based memristive elements is designed, the dynamical behaviors are revealed by theoretical analysis and numerical simulations.A. Design of memristive elements chaotic circuitThe idea of designing the memristive elements chaotic circuit is inspired by the published discussion in 14. By replacing the normal element with memory element, the memristive elements chaotic circuit can be obtained, as shown in Fig.1. By applying Kirchhoffs laws to the circuit in Fig.1, the five state equations of the memristive elements chaotic circuit are described by The fifth order dimension- less mathematical model can be obtained by system (6) where , , , , , Fig.1: memristive elements chaotic circuit (a) (b) (c) (d) (e)Fig.2: chaotic attractors of the memristive elements chaotic circuit: (a) The 3D chaotic attractor U-Y- X plane. (b) and (c) are the 2D chaotic attractor in Z-X and X-V plane respectively. (d) The 2D period limit cycle Z-X plane. (e) The 3D period limit cycle U-Y-X plane.If we set a= 0.7, a = 1.4, b = 1.4, c = 1, d = 1, e = 1, g = 1 and initial conditions (0.06, 0, 0.1, 0, 0), the novel chaotic attractors can be exhibited as shown in Figure.2(a), (b) and (c), if we set a=2 and other values are unchanged, the dynamical behaviors of the system shows a periodic state as shown in Figure.2(d) and (e).B. Equilibrium and its stabilityThrough setting the of equation (7), we can obtain the equilibrium set of the memristive elements chaotic circuit, where k1 and k2 are real constant, it signifies that any points on the y-u plane is the equilibrium points of the system. The Jacobian matrix at this equilibrium set of memristive elements circuit can be derived as where ,.Then the characteristic equation can be described as follows If k1=1 and k2=2, the five eigenvalues of the equilibrium set A is derived The equilibrium set A is an unstable saddle-focus because eigenvalues include a pair of conjugate complex root whose real part is positive.C. Dynamical behaviorsCalculations of Lyapunov exponents(LE) are widely used to indicate the existence of chaos or hyperchaos. The LE1 = 0.1773, LE2 = 0.036, LE3 = 0.0116, LE4= 0.1617, LE5 = 0.8652 and the Lyapunov dimension dL = 4.0202 can be gained by using the LET toolbox. The value of the time-varying Lyapunov exponents can be obtained by using wolfs method 27 in Fig.3, which means that the sum of these five Lyapunov exponents is negative and the maximal Lyapunov exponent is equal to 0.1773. So we conclude that the proposed chaotic circuit satisfies the condition to produce chaotic oscillation. The bifurcation diagram and Lyapunov exponent with respected a is showed in Fig.4 (a) and (b), it can be observed that the steady and unstable state of a goes along in the regions 0.01, 3.5, the system is consistent with each other. System (7) performs as a periodic behaviors when the a goes along in the regions 0.05, 0.1, 0.9, 0.96, 1.01, 1.29 and 1.31, 3.5, the first Lyapunov exponent is mainly zero. The chaotic behavior mainly happen in the regions of a goes along in the regions 0.1,0.9, 0.96, 1.01 and 1.29,1.31. These numerical simulations indicate the proposed circuit in Fig.1 can generate two different topological structure of chaotic attractors.Fig.3:Time-varying in the region 0,500 of Lyapunov exponentIV. Circuit Simulation Result In this section, the electronic circuit is simulated to verify the chaotic behavior of the system (7). The schematic of the electronic circuit as shown in Fig.5. The main elements of the circuit have including op-amp OP07CD, multiplier AD633JN, linear resistances and linear capacitors. The circuit equation (13) can be obtained according to the circuit schematic and Kirchhoffs circuit laws. (a) (b)Fig.4. (a) Bifurcation diagram and (b) Lyapunov exponent with respect to aThe main parameters are selected as R1 = R2 = R7 = R8 = R9 = R10 = R12 = R13 = R14 = R15 = R18 = 100k, R3 = R5 = 71.429k, R4 = R6 = 23.809k, R11 = R16 = 33.333k, R17 =142.857k, R19 =16.667k, R20 =11.111k, C1 = C2 = C3 = C4 = C5 = 100nF. The circuit simulation results on the Multisim platform are shown in Fig. 6(a) and (b). V. ConclusionIn this paper, a novel chaotic circuit is presented which are by using the memristor, memcapacitor and meminductor. The nonlinear dynamical behaviors are investigated by using the Phase Diagrams, equilibrium set, Lyapunov exponent, bifurcation diagram. The theoretical analyses and Fig.5 Electronic circuit diagram (a) (b)Fig.6: The circuit simulation result (a) Phase portrait in the Z-X plane. (b) Phase portrait in the V-X planenumerical simulation result shows that this circuit can produce two different dynamical behavior including chaotic oscillation and period. This system has potential application values in the fields of encryption and decryption system and neural network etc. because of complex dynamical behavior. This type of circuit will contribute to researching the physical characteristics of memristive elements and expanding nonlinear electronic circuits.Acknowledgment This work is supported by National Natural Science Foundation of China (NSFC Grant No.61550110248). The authors would like to thank the editor and the reviewers for their helpful suggestions and valuable comments.References1 L. O. Chua, “MemristorThe missing circuit element,” IEEE Trans. Circuit Theory, vol. 18, pp. 507519, Sep. 1971.2 L. O. Chua and S. M. Kang, “Memristive devices and systems,” Proceedings of the IEEE, vol. 64, no. 2, pp. 209-223, Feb. 1976.3 D. B. Strukov, G. S. Snider, D. R. Stewart, and R. S. Williams, “The missing memristor found,” Nature, vol. 453, no. 7191, pp. 8083, 2008.4 M. D Ventra, Y.V. Pershin, and L.O. Chua, “Circuit elements with memory: memristors, memcapacitors and meminductors”, Proceedings of the IEEE, no.97, vol.10, pp.1717-1724, 2009.5 C. Xu, X. Dong, N. Jouppi, and Y. 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