杭州论文 Dynamical Behaviors of A Modified Chua’s Circuit

Dynamical Behaviors of A Modified Chua s Circuit Huihui Ma Yongbin Yu Chenyu Yang Nijing Yang Yancheng Wang and Xiangxiang Wang School of Information and Software Engineering University of Electronic Science and Technology of China Chengdu China Tashi Nyima School of Information Science and Technology Tibet University Tibet China nmzx ybyu Abstract A modified Chua s circuit is implemented meminductor and the improved memristive diode bridge emulator in this paper The state equations of the modified Chua s circuit are described The dynamical behaviors such as equilibrium set Lyapunov exponents and diagram are revealed by theoretical analysis and numerical simulations Furthermore the circuit exhibits rich dynamical behaviors including chaos hyperchaos periodic windows period states crisis scenarios and coexisting attractors It is noted that coexisting attractors depend on the initial values and parameters of system Furthermore the simulation results demonstrate that some strange chaotic attractors such as double scroll type and like butterfly attractor are displayed when parameter and initial value are changing Keywords Chua s circuit Dynamic attractors simulation I INTRODUCTION HEADING 1 In 1971 Leon Chua proposed the fourth fundamental passive element which was called memristor 1 it has not been fabricated until 2008 2 In 2009 Ventra M D extended the concept of memristor to memcapacitor and meminductor 3 It is noted that memristor is a two terminal circuit element that has many unique properties such as nonvolatility 4 nonlinearity 5 and nanometer geometries 2 Hence memristor is widely applied to various fields like data storage 6 neural networks 7 8 secure communications 9 filter circuit 10 and so on In particular memristor composed circuits are more likely to generate high frequency chaotic oscillation signals Because of the controllable and nanoscale attributes of memristor it can be used to stimulate the development of the nonlinear field In recent years many nonlinear researchers are devoted to studying chaotic circuit based on memristor memcapacitor or meminductor Itoh and Chua proposed several nonlinear oscillators in 2008 11 Muthuswamy presented a simplest chaotic circuit that includes three elements of the memristor inductor and capacitor in the literature 12 Fang Yuan and Gangyi Wang proposed a meminductor based circuit in 2016 13 Bao Bocheng et al presented a simple third order memristive band pass filter chaotic 14 non autonomous seconed order memristive chaotic circuit 15 memristor based Wien bridge oscillator 16 17 and so on Literature 18 and 19 presented chaotic circuit based on meminductor and memcapacitor respectively However there are few papers that have reported chaotic circuit based on memristor and memcapacitor Furthermore many researchers have analyzed complex dynamic behaviors of chaotic circuits For example coexisting attractors were analyzed in the literature 14 20 21 state transition was reported in 22 multi hidden attractors were studied in 23 In this paper we focus on the construction of a novel Chua s circuit based on meminductor and the improve memristive diode bridge emulator The complex nonlinear dynamic behaviors such as chaos hyperchaos periodic windows period states crisis scenarios and coexisting attractors are investigated by using the Phase Diagrams equilibrium set Lyapunov exponent and bifurcation diagram It is worthy of noticing that the coexisting attractors depending the system initial value and parameters are studied The rest of this paper is organized as follows Section proposes a novel Chua s circuit based on meminductor and the improve memristive diode bridge emulator and introduces the prerequisite knowledge Section analyzes the equilibrium set of the chaotic circuit Experimental results and dynamical behavior of circuit are provided in section Finally the conclusions are drawn in section II MODELING OF THE IMPROVED CHUA S CIRCUIT The modified Chua s circuit is presented in this paper as shown in Figure 1 By replacing the Chua s diode and inductor with improved memristive diode bridge emulator and meminductor Herein the improved memristive diode bridge emulator is reported in literature 24 The circuit scheme of the improved memristive diode bridge emulator is shown in Figure 1 v and i are the input voltage and current of the improved memristor diode bridge emulator respectively iL is the current of inductor The mathematical model is described as D1 D2 D3 D4 i4 i3 i2 i1 i iL L v Figure 1 the improve memristive diode bridge emulator vM v1 v2 G L2 iM i3 i1i2 i4 C2 C1 Figure 2 The modified Chua s circuit 2 tanh SL iIiv 1 2 exp 1 ln 2cosh SL S Iivdi Lv dtIv 2 where1 2 T nV IS n and VT are the internal states of diode The smooth continuous cubic monotone increasing nonlinearity meminductor 13 is described by 3 qcd 3 The derivative equation form 4 with respect to time t can be obtained 2 i tcdt 4 where c and d are constants tddt The two models are applied in modified Chua s circuit Figure 2 shows that the modified Chua s circuit contains two capacitor C1 and C2 a negative conductor G a memristor MR and a meminductor ML By applying Kirchhoff s laws to the circuit in Fig 2 the five state equations of the modified Chua s circuit are described by 21 11 1 21 22 212 2 1 2 tanh 2 exp ln 2cosh LSL L L SLL S L dv CmnIiv dt d vv dt dv CmnGv dt Iivdi Lv dtIv d dt 5 Denote x v1 y L1 z v2 w iL u dv dtv vx y z w u a 1 C1 b IS c e 1 C2 f L2 The fifth order dimension less mathematical model can be obtained by system 5 2 2 2 2 tanh ln 2 cosh ln 2 xa mnuyabwcx yzx ze mnuyGev wbcxbwcf uy 6 When the parameters are given by a 1 b 0 005 c 1 d 1 G 1 m 3 n 7 f 0 005 and initial conditions 0 224 0 2024 0 1 0 000154 1 2294 the double scroll chaotic attractors can be exhibited as shown in Figs 3 III EQUILIBRIUM AND ITS STABILITY Through setting the 0Axyzwu of equation 6 the equilibrium set of the system can be described as 0 Ex y z w uxyzwuk 7 where k is a real constant The Jacobian matrix at this equilibrium set of a modified Chua s circuit can be derived as 2 2 2 000 10100 0 100 0001 20 01000 bmnk Jmnk bf 8 Then the characteristic equation can be described as follows 432 01234 det hhhh IJ 9 where 1 2 2 2 3 2 4 19998 99 20205 99 14 119796 97279992 93 60600 141400 h hk hk hk According to the Routh Hurwitz conditions the set of inequalities are given by 1 123 2 123143 2 4123143 0 0 0 0 h hhh h h hhhh h h h hhhh 10 If the set of inequalities 10 above are satisfied the equilibrium point is stable leading to the occurrence of point attractor Whereas if any one of the four conditions is not satisfied the equilibrium point is unstable which is the necessary condition for chaos When the other parameters are unchanged the corresponding stable regions in the parameter k is changed from 3 6048 to 26 8672 IV DYNAMICAL BEHAVIOR OF A MODIFIED CHUA S CIRCUIT A Parameter Dependent Bifurcation Behaviors With the variation of the parameter f the bifurcation diagram and first three maximum Lyapunov exponents are plotted in Figs 4 It can be observed that the chaotic circuit has rich dynamical features including chaos hyperchaos periodic windows and period states When the f goes along in the regions 0 0001 0 008 the system shows chaotic behavior the hyperchaos dynamic phenomena has occurred of f1 going along in the region 0 008 0 034 In the region of the f going along 0 034 0 048 some periodic windows with different periodicities occur Finally the system enters into period at f 0 048 Figs 5 shows a variety of phase diagrams when f takes different values Figs 5 a and Figs 5 b show the phase diagrams when system is in chaos Figs 4 c and Figs 4 d show the phase diagrams when system is in hyperchaos Figs 5 e and Figs 5 f show that the phase diagrams when system is in period a b Figs 3 chaotic attractors of a modified Chua s circuit a Y U plane b X Y plane a b Figs 4 for the initial values 0 224 0 2024 0 1 0 000154 1 2294 a Bifurcation diagram of f1 b First three Lyapunov exponents a b c d e f Figs 5 The different phase diagrams when f takes different values a X W plane when f 0 0005 b X U Y plane when the f 0 0005 c X W plane when f 0 0134 d X U Y plane when f 0 0134 e X W plane when f 0 06 f X U Y plane when f 0 06 B Multiple coexisting attractors depending on x 0 Considering that the x 0 increases in the region 2 2 the parameters of system 6 are unchanged and initial conditions are set as x 0 0 2024 0 1 0 000154 1 2294 and x 0 0 2024 0 1 0 000154 1 2294 respectively The bifurcation diagram and first three maximum Lyapunov exponents are plotted in Figs 4 The red trajectories in the Figs 6 a is when the initial value is x 0 0 2024 0 1 0 000154 1 2294 and the blue trajectories is when the initial value is x 0 0 2024 0 1 0 000154 1 2294 The corresponding three Lyapunov exponents are shown in Figs 6 b and Figs 6 c In the region 2 0 25 and 0 25 2 the system 6 displays coexisting period and strange attractors as shown in Figs 7 a and Figs 7 b However in the region 0 25 0 25 the system occurred chaotic behaviors as shown in Figs 7 c It is noted that the crisis scenarios happen at x 0 0 25 a b c Fig 6 Bifurcation diagrams of x and first three exponents with the initial x 0 increasing a the bifurcation diagrams b and c the Lyapunov exponents a b c Figs 7 the coexisting phase trajectories X Y plane a x 0 1 b x 0 1 c x 0 0 0224 V CONCLUSION In this paper a modified Chua s circuit based on meminductor and an improved memristive diode bridge emulator is implemented The nonlinear dynamic behaviors of the chaos circuit by using the chaos attractor equilibrium set Lyapunov exponent and bifurcation diagram have been analyzed Simulation result shows that this circuit have generated rich dynamical behaviors including chaos hyperchaos periodic windows period states crisis scenarios and coexisting attractors with changes in parameters and initial values Not only can this circuit be applied into the encryption system and neural network but also be used to stimulate the development of the nonlinear field 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 REFERENCES 1 L O Chua Memristor The missing circuit element IEEE Trans Circuit Theory vol 18 Sep 1971 pp 507 519 2 D B Strukov G S Snider D R Stewart and R S Williams The missing memristor found Nature vol 453 2008 pp 80 83 3 M Di Ventra Y V Pershin and L O Chua Circuit elements with memory memristors memcapacitors and meminductors Proc IEEE vol 10 2009 pp 1717 1724 doi 10 1109 JPROC 2009 2021077 4 Y Ho G M Huang and P Li Nonvolatile memristor memory device characteristics and design implications IEEE ACM International Conference on Computer aided Design digest of Technical Papers Nov 2009 pp 485 490 5 T Prodromakis BP Peh C Papavassiliou C Toumazou A Versatile Memristor Model With Nonlinear Dopant Kinetics IEEE Transactions on Electron Devices vol 58 2011 pp 3099 3105 6 P Mazumder S M Kang and R Waser Memristors Devices models and applications Proceedings of the IEEE vol 100 June 2012 pp 1911 1919 7 M Hu E raves C C Li et al Memristor Based Analog Computation and Neural Network Classification with a Dot Product Engine Advanced Materials vol 30 2018 8 B Li Y Wang Y Wang et al Training itself Mixed signal training acceleration for memristor based neural network IEEE Asia and South Pacific Design Automation Conference 2014 pp 361 366 9 J Sun Y Shen Q Yin C Xu Compound synchronization of four memristor chaotic oscillator systems and secure communication Chaos An Interdisciplinary Journal of Nonlinear Science vol 23 2013 pp 013140 10 Y Li C Yang Y Yu FF D ez Research on low pass filter based on Memristor and memcapacitor Control Conference pp 5110 5113 2017 11 Makoto Itoh L O Chua Memristor oscillators International Journal of Bifurcation and Chaos vol 18 2008 pp 3183 3206 12 B Muthuswamy Implementing memristor based chaotic circuits Int J Bifurcation Chaos vol 20 May 2010 pp 1335 1350 13 F Yuan G Wang P Jin X Wang G Ma Chaos in a Meminductor Based Circuit International Journal of Bifurcation Chaos vol 26 2016 doi 10 1142 S0218127416501303 14 B Bao N Wang Q Xu H Wu Y Hu A Simple Third Order Memristive Band Pass Filter Chaotic Circuit IEEE Transactions on Circuits Systems II Express Briefs vol 64 Aug 2016 pp 977 981 doi 10 1109 TCSII 2016 2641008 15 Q Xu Q Zhang B Bao Y Hu Non Autonomous Second Order Memristive Chaotic Circuit IEEE Access vol 5 July 2017 pp 21039 21045 doi 10 1109 ACCESS 2017 2727522 16 H Bao N Wang H Wu Z Song B Bao Bi Stability in an Improved Memristor Based Third Order Wien Bridge Oscillator IETE Technical Review vol 6 2018 pp 1 8 d