2018年论文 2018_Controlling extreme multistability of memristor emulator-based dynamical circuit in flux–charge domain

Nonlinear Dyn 2018 91 1395 1412 https doi org 10 1007 s11071 017 3952 9 ORIGINAL PAPER Controlling extreme multistability of memristor emulator based dynamical circuit in fl ux charge domain Mo Chen Mengxia Sun Bocheng Bao Huagan Wu Quan Xu Jiang Wang Received 26 July 2017 Accepted 16 November 2017 Published online 25 November 2017 Springer Science Business Media B V part of Springer Nature 2017 Abstract Memristive circuit with infi nitely many equilibrium points can exhibit thespecial phenomenon of extreme multistability whose dynamics mechanism and physical control are signifi cant issues deserving in depth investigations In this paper a control strat egy for extreme multistability exhibited in an active band pass fi lter based memristive circuit is explored in fl ux charge domain To this end an incremental fl ux charge model is established with four additional constant parameters refl ecting the initial conditions of all dynamic elements Thus the line equilibrium point only related to memristor initial condition in the voltage currentdomainistransformedintosomedeter mined equilibrium points whose locations and stabil ities are explicitly related to all four initial conditions Consequently the initial condition dependent extreme multistability phenomenon which has not been quan titatively analyzed in the voltage current domain can readily be investigated through evaluating these deter minedequilibriumpoints Mostimportantofall theini tial condition dependent dynamical behaviors are for mulated as the system parameter dependent behaviors in the newly constructed fl ux charge model and thus can be rigorously captured in a hardware equivalent realization circuit Numerical simulations and experi mentalmeasurementsrevealthatthecontrolofextreme multistability is successfully achieved in fl ux charge M Chen M Sun B Bao B H Wu Q Xu J Wang School of Information Science and Engineering Changzhou University Changzhou 213164 China e mail mervinbao domain whichissignifi cantforseekingpotentialengi neeringapplicationsofmultistablememristivecircuits Keywords Flux charge domain Memristive circuit Extreme multistability Equilibrium point Control 1 Introduction Multistability with the coexistence of multiple attrac tors is an intrinsic phenomenon of nonlinear dynami cal circuits systems whose trajectory selectively con verges on either of these coexisting attracting sets dependingonitsinitialcondition 1 22 Fornonlinear dynamical circuits systems with more than one deter mined equilibrium point a disconnected attractor may be generated surrounding each unstable equilibrium point leading to the generation of multistability Such as the traditional Chua s circuit with a pair of symmet ric unstable nonzero equilibrium points it commonly exhibits bistabililty with a pair of symmetric attrac tors or tristability with a pair of symmetric attractors and an external limit cycle with relative large ampli tude 1 When these two nonzero equilibrium points turn into stable node foci up to fi ve 8 14 or six 15 coexisting attracting sets will be evolved More spe cially for memristive circuits systems with infi nitely many equilibrium points e g line equilibrium point or plane equilibrium point infi nitely many coexisting attractorsdependingontheinitialconditionsofthestate variableshavealsobeenreported 16 18 Thisspecial 123 1396M Chen et al phenomenon characterized as extreme multistability 19 could be also found in high order coupled sys tems 20 22 A nonlinear dynamical circuit system with extreme multistability provides more freedom for the design of chaos based engineering applications 4 23 How ever in practice a proper control scheme should be applied to direct the circuit system to our desired operation mode 23 26 which still remains to be a challenge Consequently the reported multistability or extreme multistability phenomena are commonly ver ifi ed in hardware circuit through randomly triggering theexperimentalcircuit 11 17 orjustthroughnumer ical or circuit simulations 5 9 In addition for non linear dynamical circuit system with infi nitely many equilibrium points stabilities of the equilibrium points are hardly to be determined due to the infl uence of the zeroeigenvalues 16 17 27 29 Theemergingdynam ical behaviors cannot be precisely interpreted from the stabilities of the nonzero eigenvalues which are sig nifi cantly affected by the zero eigenvalues and initial conditions of dynamic elements 16 17 30 But these infl uences are hardly to be analyzed in the voltage current domain In this paper we try to investigate and control the special extreme multistability phenomenon in fl ux charge domain Flux charge analysis method was fi rst proposed as a dimensional decreasing method 30 31 in which the initial values of dynamic elements are not explicitly expressed and the special initial condition dependentbehaviorsaresurprisinglylost Thisproblem has been latterly solved by adding additional constants to represent the initial conditions 32 35 Recently a fl ux charge analysis method in terms of the incre mental fl ux and charge has been proposed 34 and adopted for investigations of parameter independent bifurcation behaviors in a memristive Chua s circuit 35 In 35 the four initial condition related system parameterswerefi nallyformulatedasasingletermand bifurcationsonafi xedmanifoldwerethennumerically analyzed Inspiredbythis iftheseinitialconditionsare separately examined in the fl ux charge domain their infl uences may readily be interpreted and the control of the sensitive initial condition dependent dynamics may become physical achievable Amemristoremulator baseddynamicalcircuitinves tigated in our previous work 17 is taken as an exam ple to explore the effi ciency of our tentative concep tion Its initial condition dependent dynamics are thor oughlyrevealedthroughbifurcationdiagrams andthen theirdynamicalmechanismsareinterpretedinthefl ux charge domain and fi nally the coexisting infi nitely many attractors are rigorously captured in a hard ware circuit In 17 according to nonzero eigenval ues of the line equilibrium point the stability distri bution related to memristor initial condition was plot ted but it was not totally matched with the exhibited initial condition dependent dynamics In addition the coexisting infi nitely many attractors were only verifi ed through circuit and numerical simulations Our current work can overcome the defi ciencies in uncovering and controlling extreme multistability It provides a new solution for the investigations of multistable circuits and systems For this end a rigorous constitutive relation of the adopted memristor emulator is formulated by integrat ing its voltage current relation over the time With thisrigorousconstitutive relation anincremental fl ux chargemodelisconstructedwithfourinitialcondition related system parameters refl ecting the infl uence of dynamic elements initial conditions In this way three main advantages are achieved 1 The initial condition dependent extreme multista bilityphenomenon observed inthevoltage current domaincanreadilybetransformedintothedynam ical behaviors relying on system parameters in the fl ux charge domain 2 The original line equilibrium point is converted into some determined equilibrium points whose locations and stabilities are evolved with the varia tion of initial condition related system parameters Thus how each individual initial condition affects the bifurcation sequence of the memristive circuit is clarifi ed by evaluating these determined equilib rium points 3 The control of extreme multistability can be phys ically achieved in the equivalent realization circuit of the incremental fl ux charge model Therestofthepaperisstructuredasfollows InSect 2 extreme multistability of the active BPF based mem ristive circuit is introduced The infl uence of the zero eigenvalue and other three initial conditions are illus tratedindetailthroughbifurcationdiagrams InSect 3 an incremental fl ux charge model of the memristive circuitisformulatedandthestabilityofthedetermined equilibrium pints is evaluated In Sect 4 with the vari ationoftheinitialconditionrelyingsystemparameters 123 Controlling extreme multistability of memristive circuit1397 Fig 1 Circuit schematic of the active BPF based memristive circuitiM R C2 C1 V11 1 R1 R4 R3 V2 R2 U 2 2 V3 C3 C0 V0 Ua Ra Rb MbMaW V0 Ub the initial condition dependent extreme multistability phenomenonisinterruptedfromtheevolutionsofthese determinedequilibriumpoints InSect 5 anequivalent realizationcircuitoftheincrementalfl ux chargemodel is constructed and experimental measurements of the initial condition dependent behaviors are performed which yields that the control of extreme multistability are physically achieved in hardware circuit The con clusions and proposals for future works are drawn in the last section 2 Active BPF based memristive circuit with extreme multistability The active BPF based memristive circuit investigated in 17 is taken as a paradigm for our investigations The circuit schematic is given in Fig 1 where an ideal active voltage controlled memristor emulator is used The circuit parameters are fi xed as Ra 10k Rb 1 4k R 1 5k R1 100 R2 10k R3 1k R4 50 C0 C1 5nF C2 C3 100nF and g 0 1 Note g is the total gain of the multipliers Maand Mb Commonly the memristive circuit in Fig 1 is mod eled in the voltage current domain and four fi rst order autonomousdifferentialequationsintermsoffournode voltages of V0 V1 V2 and V3are obtained as C0dV0 dt 1 RaV1 C1dV1 dt 1 RV1 1 RV2 1 Rb gagb Rb V2 0 V1 CdV2 dt k RV1 k R V2 2k 1 k 1 R2V3 CdV3 dt k 1 R V1 k 1 R V2 2 R2V3 1 Fig 2 Eigenvalues of the line equilibrium point P in the region of 2 2 2 2 where i 1 2 3 4 where V0is the inner state variable of the memristor emulator R RR1 R R1 and k R4 R3 Thiscircuithasalineequilibrium whichisexpressed as P V0 V1 V2 V3 V1 V2 V3 0 V V0 V 2 where the constant is uncertain Based on the Jaco bian matrix of 1 17 when the parameter is increased from 2 2 to 2 2 four eigenvalues at the line equilibrium P are calculated via MATLAB numerical simulations and the real parts of the obtained eigen values are plotted in Fig 2 Obviously there are a zero root 1 arealroot 2 andapairofconjugatedcomplex roots 3and 4 From Fig 2 it can be found that the stability dis tributions of the nonzero eigenvalues in the negative parameter space are symmetric to those in the pos itive parameter space and the nonzero equilibrium points are stable in the region of 1 119 1 483 However it has been revealed that the stability of the 123 1398M Chen et al Fig 3 Bifurcation diagrams of V3 t with increasing from 2 2 to 2 2 the initial conditions are set as a V 10 9V 0 V 0 V b V 0 15 V 0 V 0 V c V 0 V 0 05 V 0 V and d V 0 V 0 1 V 0 1 V active BPF based memristive circuit can not be sim ply determined by the three nonzero eigenvalues of the line equilibrium 17 Since 1always equals to zero the stability of P cannot be explicitly distinguished just with respect to these nonzero eigenvalues which leads to some diffi culties in the predictions of the gen erated dynamical behaviors 16 17 30 Besides the initial conditions of other three dynamic elements C1 C2and C3 also have great infl uences on the dynamical behaviors of the active BPF based memristive circuit To illustrate this problem bifurcation diagrams of the state variable V3 t with the increasing of V0 0 are simulated and presented in Fig 3 where the initial conditions V0 0 V1 0 V2 0 V3 0 arechosenas V 10 9V 0 V 0 V V 0 15 V 0 V 0 V V 0 V 0 05 V 0 V and V 0 V 0 1 V 0 1 V respectively Apparently with the variation of V0 0 i e the memristive circuit exhibits different dynamical behav iors leadingtothegenerationofextrememultistability Buttheexhibitedbifurcationsequencesdonotconform to the stability distributions presented in Fig 2 and are greatlychangedwiththevariationsof V1 0 V2 0 and V3 0 Numerical simulation results given in Fig 2 illus trate the interesting phenomena of extreme multista bility but mathematical analyses are still desired to 123 Controlling extreme multistability of memristive circuit1399 provide more general understanding of this special phenomenon But unfortunately the infl uence of the zeroeigenvalue andtheinitialconditionsofotherthree dynamicelementscannotbeformulatedinthevoltage current domain 3 Incremental fl ux charge model of the memristive circuit Generally speaking the dynamical behaviors of a nonlinear system are commonly characterized by its equilibrium points 36 38 but for the systems with infi nitely many equilibrium points stabilities of their equilibrium points are hardly to be determined due to theinfl uenceofthezeroeigenvalues 16 17 27 29 To get better insight into the initial condition dependent extreme multistability phenomenon we try to con vert the voltage current model 1 into the fl ux charge model inwhichthefourinitialconditionsV0 0 V1 0 V2 0 and V3 0 are formulated as four standalone system parameters and the original line equilibrium point is transformed into some determined equilibrium points These benefi ts can facilitate the lucubration on the dynamical mechanism of extreme multistability 3 1 Rigorous constitutive relation of the memristor emulator Themathematicalmodelofthememristoremulatorcan be described as 17 iM W V0 V1 1 Rb 1 gV2 0 t V1 t 3 dV0 dt f V1 1 RaC0 V1 t 4 Its incremental charge within the time interval of 0 t is formulated as qM t 0 t 0 1 Rb 1 gV2 0 V1 d t 0 1 Rb 1 gV2 0 RaC0dV0 1 Rb V0 t V0 0 g 3 Rb V3 0 t V 3 0 0 5 where 1 RaC0 Based on 4 we have V0 t 1 RaC0 t 0 V1 d V0 0 1 t 0 V0 0 6 Put 6 into 5 a rigorous constitutive relation of the memristor emulator is obtained as qM t 0 g 2 3Rb 1 t 0 3 g Rb V0 0 1 t 0 2 1 Rb 1 gV0 0 2 1 t 0 7 In 7 1 t 0 is the incremental fl ux of capacitor C1 and V0 0 is the inner initial value of the memristor emulator Unlike its original ideal constitutive relation 17 theinitialconditionofitsinnerstatevariableV0 t isexplicitlyexpressed whichisofgreatimportancefor the analyses of extreme multistability caused by the memristor initial condition 3 2 Construction of the incremental fl ux charge model Suppose the circuit is powered on at t 0 V1 0 V2 0 and V3 0 denote the initial voltages of three capacitors and 1 t 0 2 t 0 and 3 t 0 are the incremental fl ux of capacitors C1 C2and C3 respec tively Then the incremental fl ux charge model can be obtained by integrating 1 from 0 to t For the second equation of 1 there is V1 t V1 0 1 RC1 1 t 0 1 RC1 2 t 0 1 C1 qM t 0 8 Since the fl ux of capacitor C1is 1 t t V1 d t 0 V1 d 0 V1 d 1 t 0 Constant 9 V1 t can be rewritten as V1 t d 1 t dt d 1 t 0 dt 10 and 8 can be formulated in terms of the incremental fl ux and charge as d 1 t 0 dt 1 RC1 1 t 0 1 RC1 2 t 0 1 C1 qM t 0 V1 0 11 123 1400M Chen et al Similarly based on the third and fourth equations of 1 we have d 2 t 0 dt k RC 1 t 0 k R C 2 t 0 2k 1 k 1 R2C 3 t 0 V2 0 12 d 3 t 0 dt k 1 RC 1 t 0 k 1 R C 2 t 0 2 R2C 3 t 0 V3 0 13 Finally a dimensionality reduction fl ux charge model is deduced as d 1 t 0 dt 1 RC1 1 t 0 1 RC1 2 t 0 1 C1qM t 0 V1 0 d 2 t 0 dt k RC 1 t 0 k R C 2 t 0 2k 1 k 1 R2C 3 t 0 V2 0 d 3 t 0 dt k 1 RC 1 t 0 k 1 R C 2 t 0 2 R2C 3 t 0 V3 0 14 It should be noted that the initial conditions of 14 are fi xed as 1 0 0 0 2 0 0 0 3 0 0 0 15 which ensures that the dynamics uncovering from 14 is identical to that revealed from 1 To simplify the analysis procedure three new vari ables are introduced and the circuit parameters are scaled in a dimensionless form as x 1 t 0 RC y 2 t 0 RC z 3 t 0 RC t RC u du d u x y z a1 g 2R3C2 3Rb a2 g R2C Rb a3 gR Rb a4 R Rb b C C1 r1 R R1 r2 R R2 k R4 R3 1 RaC0 0 V0 0 1 V1 0 2 V2 0 3 V3 0 16 Accordingly 14 can be written as x bh x by 1 y kx k 1 r1 y 2k 1 k 1 r2z 2 z k 1 x k 1 1 r1 y 2r2z 3 17 where h x a1x3 a2 0 x2 a3 2 0 a4 1 x 18 Systemparameter 0denotestheinnerstatevariableof the memristor emulator and 1 2 and 3are system parametersrefl ectingtheinducedinitialvaluesofthree capacitors C1 C2 and C3 Specially it needs to be emphasized that the initial conditions of 17 are fi xed as x 0 0 y 0 0 and z 0 0 For the specifi ed circuit parameters used in Fig 2 the normalized parameters defi ned in 16 are calcu lated as a1 0 3214 a2 0 3214 a3 0 1071 a4 1 0714 b 20 r1 15 r2 0 15 k 0 05 2 104 19 which are kept unchanged for our next investigations Thentheinitialcondition dependentdynamicalbehav iorsofthememristivecircuitareexhibitedbychanging system parameters 0 1 2 and 3 3 3 Equilibrium point and stability Set the left side of 17 equal to zero its equilibrium points are obtained as E x h x 1 20 1 7 2 1 3 3 20 where x is numerically solved by a1x3 a2 0 x2 a3 2 0 a4 r1 1 r1 x 1 b 1 2 2 1 r1 2k 1 k 1 1 r1 3 0 21 whose roots can be expressed as x1 3 q 2 3 q 2 a2 3a1 0 22 x2 1 j 3 2 3 q 2 1 j 3 2 3 q 2 a2 3a1 0 23 x3 1 j 3 2 3 q 2 123 Controlling extreme multistability of memristive circuit1401 1 j 3 2 3 q 2 a2 3a1 0 24 In 22 24 p and q are defi ned as q2 4 p3 27 p a 2 2 2 0 3a2 1 a3 2 0 a4 1 a1 1 a1 1 r1 q 2a3 2 3 0 27a3 1 a2a3 3 0 a2a4 0 a2 0 3a2 1 a2 0 3a2 1 1 r1 1 a1b 1 2 a1 1 r1 2 2k 1 a1 1 r1 1 k 3 The Jacobian matrix at equilibrium point E is derived as J bh x b0 kk 1 r1 2k 1 k 1 r2 k 1 k 1 1 r1 2r2 25 where h x 3a 1 x2 2a2 0 x a3 20 a4 1 26 Thus threeeigenvalues of E areyieldedbysolvingthe following characteristic equation F det 1 J 3 m1 2 m2 m3 0 27 in which m1 bh x k 1 r1 2r2 m2 bh x 2r2 k kr1 1 r1 r2 bk m3 r2bh x 1 r1 br2 AccordingtoRouth Hurwitzconditions wecandeter mine that when x satisfi es 0 3727 x 1 3 0 0 4946 28 the corresponding equilibrium point E is stable and a point attractor will be obtained in its neighborhood Apparently inthefl ux chargedomain thelineequi librium point of 1 is converted into some deter mined equilibrium points whose locations and stabil ities are determined by the initial condition dependent system parameters 0 1 2 and 3 Thus the com plex initial condition dependent extreme multistabil ity phenomenon presented in Fig 3 can readily be