论文集 Extreme_multistability_in_a_memristor-based_multi- Extreme multistability in a memristor-based multi-scroll hyper-chaotic system

Extreme multistability in a memristor based multi scroll hyper chaotic system Fang Yuan Guangyi Wang and Xiaowei Wang Citation Chaos 26 073107 2016 doi 10 1063 1 4958296 View online http dx doi org 10 1063 1 4958296 View Table of Contents http scitation aip org content aip journal chaos 26 7 ver pdfcov Published by the AIP Publishing Articles you may be interested in Periodicity chaos and multiple attractors in a memristor based Shinriki s circuit Chaos 25 103126 2015 10 1063 1 4934653 Experimental dynamical characterization of five autonomous chaotic oscillators with tunable series resistance Chaos 24 033110 2014 10 1063 1 4890530 Chaotic saddles in nonlinear modulational interactions in a plasma Phys Plasmas 19 112303 2012 10 1063 1 4766472 A chaotic circuit based on Hewlett Packard memristor Chaos 22 023136 2012 10 1063 1 4729135 Lyapunov exponent diagrams of a 4 dimensional Chua system Chaos 21 033105 2011 10 1063 1 3615232 Reuse of AIP Publishing content is subject to the terms at https publishing aip org authors rights and permissions Downloaded to IP 115 200 234 185 On Sun 17 Jul 2016 01 46 16 Extreme multistability in a memristor based multi scroll hyper chaotic system Fang Yuan 1 a Guangyi Wang 1 b c and Xiaowei Wang2 c 1Institute of Modern Circuits and Intelligent Information Hangzhou Dianzi University Hangzhou 310018 China 2Department of Automation Shanghai University Shanghai 200072 China Received 6 April 2016 accepted 24 June 2016 published online 13 July 2016 In this paper a new memristor based multi scroll hyper chaotic system is designed The proposed memristor based system possesses multiple complex dynamic behaviors compared with other chaotic systems Various coexisting attractors and hidden coexisting attractors are observed in this system which means extreme multistability arises Besides by adjusting parameters of the system this chaotic system can perform single scroll attractors double scroll attractors and four scroll attractors Basic dynamic characteristics of the system are investigated including equilibrium points and stability bifurcation diagrams Lyapunov exponents and so on In addition the presented system is also realized by an analog circuit to confi rm the correction of the numerical simulations Published by AIP Publishing http dx doi org 10 1063 1 4958296 The memristor is the fourth basic circuit element Because of the special electrical properties a real memristor or memristor emulators are usually used to construct chaotic circuit whereas a pure mathematical model of a memristor can also be employed to design cha os system Although some novel dynamic characteristics are found in memristor based circuits extreme multi stability is rarely observed or reported which means infi nitely many attractors coexist for the same set of system parameters In this paper a memristor based system is designed for generating multiple scrolls Various hidden attractors and coexisting attractors are investigated Compared with other memristor based systems the pre sented system owns more complex dynamic behaviors including hyper chaos state controllable multiple scrolls transient period coexisting hidden attractors sustained chaos state and so on All these complex properties are just refl ections of an extreme multistability I INTRODUCTION The memristors that had been predicted as the fourth basic circuit element by Chua1in 1971 were fabricated by HP Hewlett Packard lab in 2008 2Due to the realizations and particular properties of real memristors much attention has been paid to memristor based applications The nonli nearities of memristors are utilized to construct chaotic circuits design fl ash memory and improve neural networks and so on In earlier researches memristors are usually employed to construct chaotic oscillators Reference 3 derives several nonlinear oscillators from Chua s oscillators by replacing Chua s diodes with memristors A chaotic circuit based on the mathematical realistic model of the HP memristor is introduced in Ref 4 and a Twin T notch fi lter feedback con troller is designed and employed to control the chaotic be havior in the memristor based chaotic circuit in Ref 5 Besides Ref 6 realizes a memristor using off the shelf com ponents and then constructs the memristor along with the associated chaotic circuit on a breadboard Since HP memristors are not commercially available until now researchers have been committed to develop other new kinds of memristors or emulators in recent years Reference 7 reports the development of an Ag WO3 ITO thin fi lm memristor device using a spray pyrolysis method The structural morphological and electrical properties of the thin fi lm memristor device are further characterized using x ray diffraction scanning electron microscopy and semi conductor device analyzer Similarly the development of the Ag ZnO FTO memristor device using a simple aqueous chemical route is reported in Ref 8 Besides more and more memristor based circuits and applications are presented recently A memristor based neural crossbar which imple ments on chip supervised learning and a compact learning cell that consists of a crossbar latch of two antiparallel ori ented binary memristors are presented in Ref 9 Reference 10 proposes a new twin crossbar architecture of binary mem ristors for low power image recognition in which two identi cal memristor arrays instead of the previous complementary memristor arrays of M and M Reference 11 presents a novel read write circuit that facilitates the reading and writ ing operation of the memristor device as a memory element Furthermore Ref 12 introduces a novel complex Lorenz system with a fl ux controlled memristor and realizes its synchronization Reference 13 designs a novel memristor based oscillator which is obtained from Shinriki s circuit by substituting the nonlinear positive conductance with a fi rst order memristive diode bridge Reference 14 presents experi mental analysis of chaotic dynamics in a variable memristor based Chua s Circuit In Ref 15 a periodically forced a E mail yf210yf b E mail wanggyi c G Wang and X Wang contributed equally to this work 1054 1500 2016 26 7 073107 13 30 00Published by AIP Publishing 26 073107 1 CHAOS 26 073107 2016 Reuse of AIP Publishing content is subject to the terms at https publishing aip org authors rights and permissions Downloaded to IP 115 200 234 185 On Sun 17 Jul 2016 01 46 16 memristive Chua s circuit is studied and its complex tran sient dynamics are analyzed Recently multistabilityandextrememultistability become a research focus Multistability is a common phe nomenon in many nonlinear dynamical systems correspond ing to the coexistence of more than one stable attractor for the same set of system parameters 16 When infi nitely many attractors coexist for the same set of system parameters this is called extreme multistability 17Various coexisting attrac tors and hidden coexisting attractors are essentially extreme multistability Thus many researches were reported with respect to the special dynamic behaviors of coexisting attrac tors over the past few years Reference 18 proposes a new 3D autonomous quadratic chaotic system as a typical exam ple with the presence of the coexisting attractors Reference 19 proposes a novel chaotic attractor with a fractal wing structure including two coexisting periodic or strange attrac torsthatcancoexistwithathirdstrangeattractor Furthermore hidden attractors are reported to fi nd in a nor malized Chua s equation in Ref 20 Coexisting attractors and hidden attractors are also observed in a generalized memristor based Chua s circuit 21 To our best knowledge the researches of the extreme mul tistability in memristive circuits are still relatively less In this paper the mathematical model of memristor is employed to construct multi scroll chaos system Distinguishing from most other chaos systems the presented memristor based system displays more complex dynamic characteristics including coexisting attractors hidden coexisting attractors constant Lyapunov exponents and controllable numbers of scroll and so on The rest of this paper is organized as follows In Section II the model of memristor and its equivalent circuit are intro duced Besides a new hyper chaotic system based on the mod el of memristor is studied In Section III multiple dynamical characteristics of the system are discussed In Section IV the presented chaotic system is realized by artifi cial circuit and the experimental results are given Finally some conclusions are drawn from the present study in Section V II MEMRISTOR MODEL AND A MEMRISTOR BASED HYPER CHAOTIC SYSTEM A Memristor model and its equivalent circuit A generalized nth order memristor is presented in Ref 22 which is governed by the following relation VM M q I M q du q dq 8 1 I W u V W u dq u du 8 2 where M q indicates memristance and W u indicates mem ductance The symbol q and u are the charge and fl ux across the memristor respectively The smooth fl ux controlled memristor model is widely researched 23 25which is described as W u dq u du a bu2 3 where a and b are constant coeffi cients The parameter a indicates the polarity of the memristor In the case of a 0 the memristor is passive and a 5 where x y z and w are state variables and a b c and k are parameters The function W w represents a fl ux controlled memristor mathematical model that can be depicted as W w a bw2 6 If we set parameters as in Table I and initial condition as 0 01 0 0 0 hyper chaotic attractors can be obtained as shown in Fig 4 where four scroll attractors are displayed in x y x z and y w phase portraits and hyper chaotic attractor in z w phase portrait that is limited by two parallel lines in the region of 15 15 In this case the corresponding Lyapunov exponentsarecalculatedasLE1 2 576 LE2 0 214 LE3 0 34 LE4 14 12 and the Lyapunov dimension is dL 3 1735 which indicates the system is hyperchaos Furthermore the corresponding Poincar e section on z 0 in the region of k 2 30 30 is as shown in Fig 5 a and b 2 0 3 22 in Fig 5 b which also illustrates that this hyper chaotic system has complex dynamic behaviors III DYNAMICAL BEHAVIORS OF THE SYSTEM A Equilibrium points The equilibrium state of system 5 can be obtained by solving the equations x y z w 0 There are ten equilibrium points which can be described as S1 2 0 0 0 6 ffi ffi ffi ffi ffi ffi ffiffi a b r S3 4 ffi ffi ffi ffiffi bc p ffi ffi ffi ffiffi ac p ffi ffi ffi ffi ffi ab p 6 ffi ffi ffi ffi ffi ffi ffiffi a b r S5 6 ffi ffi ffi ffi ffi bc p ffi ffi ffi ffi ffi ac p ffi ffi ffi ffi ffi ab p 6 ffi ffi ffi ffi ffi ffi ffiffi a b r S7 8 ffi ffi ffi ffiffi bc p ffi ffi ffi ffiffi ac p ffi ffi ffi ffi ffi ab p 6 ffi ffi ffi ffi ffi ffi ffiffi a b r S9 10 ffi ffi ffi ffi ffi bc p ffi ffi ffi ffi ffi ac p ffi ffi ffi ffi ffi ab p 6 ffi ffi ffi ffi ffi ffi ffiffi a b r 8 7 The corresponding Jacobian matrix J is given as FIG 2 The equivalent circuit of the memristor characterized in Eq 3 FIG 3 Multisim simulations to emu late v I hysteresis loop of the memris tor in conditions of a V1 1V and b V1 1V TABLE I Circuit parameters for hyper chaos ParametersSignificationsValues aVariable0 45 bVariable9 cVariable5 4 kVariable15 aVariable 14 bVariable0 06 073107 3Yuan Wang and WangChaos 26 073107 2016 Reuse of AIP Publishing content is subject to the terms at https publishing aip org authors rights and permissions Downloaded to IP 115 200 234 185 On Sun 17 Jul 2016 01 46 16 J a z y0 z bx2bw yx c0 0 k a bw2 0 2bkwy 8 Thentheeigenvaluesattheseequilibriumpoints Si i 1 2 3 10 are yielded by solving the following char acteristic equation det 1k J 0 9 If we keep parameters a and b as variables and set other parameters in Table I the characteristic equation at equilibri um points S1 2 can be simplifi ed as k k3 b a 6 k2 6b 6a ab k 6ab 0 10 It is obvious that the coeffi cient of cubic polynomial equation in Eq 10 is nonzero value Thus according to the Routh Hurwitz condition the necessary and suffi cient condi tions are given by Eq 11 which means the root s real parts of this polynomial are negative b a 6 0 6b 6a ab 0 6ab 0 b a 6 6b 6a ab 6ab 0 8 11 If we set a 2 0 6 and b 2 0 40 the region satisfying the conditions of Eq 11 is shown in Fig 6 where the yellow color indicates the stable region and the blue color indicates the unstable region If we set a 0 45 and b 9 the ten eigen values at these equilibrium points are calculated as in Table II FIG 4 Phase portraits of the proposed hyper chaotic system in a x y b x z c y w and d z w planes FIG 5 Poincar e sections on z 0 in regions of a k 2 30 30 and b b 2 0 3 22 073107 4Yuan Wang and WangChaos 26 073107 2016 Reuse of AIP Publishing content is subject to the terms at https publishing aip org authors rights and permissions Downloaded to IP 115 200 234 185 On Sun 17 Jul 2016 01 46 16 B Lyapunov spectra and bifurcation diagrams of hyper chaos In order to investigate the rich dynamical behaviors of the proposed hyper chaotic system bifurcation diagrams and Lyapunov spectra are used in this section If we keep b varying and set other parameters as in Table I the bifurcation diagram of the state variable x is shown in Fig 7 a with initial conditions 0 01 0 0 0 The corresponding Lyapunov exponent spectrum is described in Fig 7 b From the Lyapunov exponent spectrum it is obvi ous that the system almost keeps chaotic state in the regions of b 2 0 10 apart from several period windows nearby b 1 2 and b 5 2 In the regions of b 2 6 10 there exists two evident positive Lyapunov exponents that means the system is hyper chaotic In general the main change regular ity of the system is transforming between chaos and hyper chaos along with the variable b varying Some further exam ples of typical attractors are described in Fig 8 In particular Figs 8 a and 8 c show the projections of some different limit cycles on the x y phase plane and the projections of hyper chaotic and chaotic attractors generated by the system are shown in Figs 8 b and 8 d C Quantity control of multiple scrolls The presented hyper chaotic system not only can gener ate multi scroll chaotic attractors but also can produce single scroll attractors Simply adjusting parameters set as in Table III four kinds of single scroll coexisting chaotic attractors can be obtained by system 5 which are shown in Fig 9 The mentioned single scroll and multi scroll attractors here are aimed at x y phase diagrams In detail for coexisting attractors in Fig 9 the red one starts from the initial condi tions 1 05 0 0 0 the blue one starts from 1 05 0 0 0 the black one starts from 0 05 0 0 0 and the yellow one starts from 0 05 0 0 0 Furthermore under another parameter combination as in Table IV new kinds of twin scroll chaotic attractors can be found distinguishing with the one displayed in Fig 8 b which are described in Fig 10 The two twin scroll attractors are symmetrical with each other that start from initial condi tions 1 5 0 1 0 and 0 01 0 0 0 respectively It is obvi ous that the symmetry of the two coexisting attractors has nothing to do with the symmetry of the system equation since the corresponding two initial conditions are not sym metrical Therefore this multistability in dynamic behaviors belongs to the system itself D Symmetry and transient period in coexisting attractors Obviously the system 5 is invariant if we do the trans formation x y z w x y z w which means that x y z w is also a solution of the system for the same parameters set Thus the x z phase diagrams obtained by the conditions of x y z w and x y z w have to be sym metry by inversion This symmetry characteristic could be served to explain the presence of symmetrical coexisting attractors in state space FIG 6 The stable and unstable regions with the conditions a and b varying TABLE II Different eigenvalues ki i 1 2 3 4 of the equilibrium set Sik1k2k3k4Stability S1 20 9 5 40 45Unstable saddle point S3 5 7 942 861 14 3730 19239 2 3935i0 19239 2 3935iUnstable focal point S4 6 8 10 42 861 14 3730 21172 2 4579i0 21172 2 4579iUnstable saddle focus point FIG 7 The dynamical behavior for parameters listed in Table I along with b varying in initial conditions of 0 01 0 0 0 a the bifurcation diagram and b theLyapunovexponents obtained by system 5 073107 5Yuan Wang and WangChaos 26 073107 2016 Reuse of AIP Publishing content is subject to the terms at https publishing aip org authors rights and permissions Downloaded to IP 115 200 234 185 On Sun 17 Jul 2016 01 46 16 If we set parameters as in Table V various symmetrical coexisting attractors can be shown in Fig 11 and the corre sponding conditions are listed in Table VI The main coexist ing regimes are symmetric pairs of chaotic attractors coexisting with each other The distinction of various pairs of coexisting attractors is the different numbers of impact waves The orbits of the system and the long run dynamics depend on the starting conditions Through the method of al tering starting points the system 5 can generate different coexisting attractors which owns various impact waves as shown in Fig 11 In detail the attractors in Fig 11 a are the normal chaotic attractors When the initial conditions are al tered the chaotic attractors can jump from the chaos orbit into the impact wave orbit After quasi periodic movement in impact wave orbit the attractors return to chaos orbit again In the other words there exist odd dynamic behaviors in this system which means that the system can change from quasi periodic state to chaotic state The quasi periodic state is transient so we can call this phenomenon as transient period The huge region of attraction of the system can ex plain this transient period Since the region of attraction is huge or big enough no matter where is the starting point of the attractor it always can be drawn into a special orbit In this case the special orbit is chaotic while the process of at traction is quasi periodic E Hidden coexisting attractors Further exploring the pres