2017年论文 2017_Numerical analyses and experimental validations of coexisting multiple attractors in Hopfield neural network

123 Nonlinear Dynamics An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems ISSN 0924 090X Volume 90 Number 4 Nonlinear Dyn 2017 90 2359 2369 DOI 10 1007 s11071 017 3808 3 Numerical analyses and experimental validations of coexisting multiple attractors in Hopfield neural network Bocheng Bao Hui Qian Jiang Wang Quan Xu Mo Chen Huagan Wu particularly thissystem exhibits some striking phenomena of coexisting mul tiple attractors such as a pair of single scroll chaotic attractors accompanied with a pair of periodic attrac tors apairofperiodicattractorswithtwoperiodicities andsoon Ofparticularinterest itshouldbehighlysig nifi cant that a hardware circuit of the HNN based sys temisdevelopedbyusingcommerciallyavailableelec tronic components and many kinds of coexisting mul tiple attractors are captured from the hardware exper iments The results of the experimental measurements havewellconsistencytothoseofMATLABandPSpice simulations B Bao B H Qian J Wang Q Xu M Chen H Wu School of Information Science and Engineering Changzhou University Changzhou 213164 China e mail mervinbao Y Yu School of Mathematics and Physics Changzhou University Changzhou 213164 China Keywords Hopfi eld neural network HNN based system Coexisting multiple attractors State initial value Hardware experiment 1 Introduction For a long time Hopfi eld neural network HNN has been widely studied and applied in science and engineering such as information processing charac ter recognition data storage and associative memory stereo image matching image encryption and so on 1 6 Due to the extraordinary nonlinearity of neuron activationfunction theHNN basedsystemisastrongly nonlinear dynamical system 1 Like as the classical nonlinear dynamical systems Chua s chaotic circuits 7 9 chaotic and hyperchaotic time delay systems 10 13 and coupled HR neuron circuits 14 for just a few examples the HNN based system can generate complex dynamical behaviors of chaos hyperchaos period and quasi period 15 23 implying that the HNN basedsystemcanactuallysimulatetheemerging chaoticbehaviorsasthebrainisininformationprocess ing 24 25 Therefore the investigation of these non lineardynamicalbehaviorshassignifi cantlytheoretical meaning and comprehensively practical applications The coexisting phenomenon of somedifferenttypes of disconnected attractors is a fantastic nonlinear phe nomenon observed in nonlinear physical and mathe matical systems such as Chua s chaotic circuits 8 9 memristor based chaotic circuits 26 29 relay sys 123 Author s personal copy 2360B Bao et al tem with hysteresis and multilevel DC DC converter 30 31 recurrent neural networks 32 33 pure non linear mathematical systems 34 38 and so on The coexisting phenomenon of multiple attractors reveals the essential feature of multistability in a nonlinear dynamicalsystem 39 whichhasreceivedmuchatten tion of the investigators in recent years 39 43 Coex isting multiple attractors or multistability embodies a diversity of many steady states in a nonlinear dynami calsystem whichcanmakethesystemofferawonder fulfl exibility 39 suitableforinformationengineering applications 43 Up to now however coexisting mul tipleattractors behaviorormultistabilityphenomenon in the HNN based nonlinear system has not been pre viously reported The investigation of such a specifi c behavior in the HNN based nonlinear system is con ducive to deeply understand the dynamical effect of multistability on the aspects of brain information pro cessing and cognitive function In this paper by removing the self connection synaptic weight of the second neuron to simplify the HNN connection topology presented in 15 a novel HNN based nonlinear system is proposed from which the striking phenomena of coexisting multiple attrac tors are found by MATLAB numerical simulations and hardwareexperiments Mostofall aphysicalhardware circuit of the HNN based nonlinear system is devel oped by using commercially available electronic com ponents and many kinds of coexisting phenomena of multipleattractorsaresimulatedbyPSpicecircuitsim ulations and captured from the hardware experiments Thus ourresearchresultsarecompletelydifferentfrom those of the previous literatures only confi ned to MAT LAB and PSpice simulations 15 23 44 45 2 HNN based nonlinear system 2 1 The proposed mathematical model A new kind of Hopfi eld neural network with three neurons is presented and its connection topology is giveninFig 1 Comparedwiththeconnectiontopology reported in 15 there is no self connection synaptic weight in the second neuron of Fig 1 Thus the math ematical model for the HNN based nonlinear system is a simplifi ed model which leads to the change of the corresponding dynamical behavior 7 Neuron 1 Neuron 2 Neuron 3 k 1 11 2 2 2 8 1 4 4 Fig 1 Connection topology for the Hopfi eld neural network HNN considered The newly proposed HNN based nonlinear system can be expressed as x x W tanh x 1 where x x1 x2 x3 T R3is the neuron state vec tor tanh x tanh x1 tanh x2 tanh x3 Tis the nonlinear neuron activation function and W wij is the 3 3 synaptic weight matrix For the connection topology given in Fig 1 the connection matrix can be yielded as W w11w21w31 w12w22w32 w13w23w33 1 41 2 7 1 102 8 k 24 2 where k is the coupling connection synaptic weight betweenthefi rstandthirdneurons Theparameterk isa uniqueadjustableparameteroftheHNN basednonlin earsystem Bythefollowingresultsoftheoreticalanal yses and numerical simulations the adjustable param eter range is set to 0 k 1 34 otherwise they are a pair of unstable saddle foci when 0 536 k 1 34 Take k 0 95 as an example to further illustrate the equilibrium points and their stability of the system 1 The newly proposed system has three equilibrium points which are given as x0 0 0 0 T x 1 4734 0 1962 0 4522 T 7 respectively where x0is a zero point i e an origi nal point and x are two symmetrical nonzero points The eigenvalues of the three equilibrium points can be solved as x0 0 4887 0 4444 j1 8349 x 0 3984 j1 3322 0 7804 8 respectively whichimplythattheoriginalpoint x0isan unstablesaddlepoint whereas x areapairofunstable saddle foci Thus the system 1 may generate chaotic attractor at this determined parameter 3 Dynamics with coexisting bifurcation modes Take the coupling connection synaptic weight k as an adjustable parameter and consider the state initial val uesas 0 0 01 0 and 0 0 01 0 respectively Forthe 123 Author s personal copy 2362B Bao et al Fig 2 Dynamics of the HNN based system with respect to k a fi nite time Lyapunov exponent spectra b bifurcation diagrams of the state variable x1 k Lyapunov exponents L1 L2 L3 k x1 0 0 01 0 0 0 01 0 a b system described by 1 fi nite time Lyapunov expo nent spectra via Wolf s method 46 and bifurcation diagramsofthestatevariablex1areplottedinFig 2a b respectively where dynamical behaviors together with coexisting bifurcation modes are observed in Fig 2b the reds are the orbits starting from the state initial values 0 0 01 0 and the blues are the orbits starting from 0 0 01 0 AsshowninFig 2 theHNN based system exhibits some conventional dynamics of point period and chaos forward period doubling bifurca tion reverse period doubling bifurcation and tangent bifurcation AccordingtodynamicsgiveninFig 2andnumerical simulations of the phase plane trajectories it is known that the system 1 locates in a stable state and mani festscoexistingpointattractorswhenk 0 536 Asthe adjustable parameter k increases gradually the system 1 exhibits coexisting period 1 attractors fi rstly and thenpresentscoexistingperiod 2attractorsviaforward period doubling bifurcation route at k 0 603 Fur ther thedynamicsofthesystem 1 turnsintotherange of coexisting single scroll chaotic attractors and goes intotherangeofdouble scrollchaoticattractorafterthe coexisting bifurcation mode disappears at k 0 617 In the range of 0 613 k 0 965 chaotic behav iors mingled with several periodic windows and tan gent bifurcation routes emerge in the system 1 Like wise in the range of 0 878 k 1 343and its coexisting point attractors are merged into a point attractor at k 1 43 When k 0 52 0 75 0 8 0 95 0 97 and 1 1 are considered phase plane trajectories of the system 1 in the x1 x3plane are numerically simulated in Fig 3 where the red and blue orbits correspond to the state initial values 0 0 01 0 and 0 0 01 0 respec tively Figure 3a is coexisting point attractors imply ing that the operating orbits starting from two different stateinitialvaluestendtotwostableequilibriumpoints fi nally Figure 3b c is a period 3 attractor in a wide periodic window and a double scroll chaotic attractor respectively Figure 3d demonstrates that the system 1 generates a pair of single scroll chaotic attractors symmetricalwithrespecttotheoriginalpoint andtheir operating orbits move around two nonzero equilibrium points respectively Whereas Fig 3e f is a pair of period 4 attractors and a pair of period 1 attractors which are all symmetrical with respect to the original point As a result with the variation of the adjustable parameter dynamicalbehaviorsofstablepoint period chaos period doubling bifurcation and tangent bifur cation indeed exist in the HNN based system 4 Coexisting multiple attractors behaviors Multistability indicates the coexisting phenomenon of many different kinds of disconnected attractors in a nonlinear dynamical system which is a kind of widespread physical phenomena 8 9 26 43 For the same set of system parameters with different state ini tial values the system orbits may asymptotically tend to different stable states of point chaos period and quasi period Under different state initial values the HNN based system can show several kinds of coexisting multi ple attractors behaviors in some neighborhoods of the parameterk Consideringthreerangesoftheparameter values near k 0 95 0 9 and 0 67 as examples the coexisting phenomenon of multiple attractors of the system 1 is elaborated by bifurcation diagrams and phase plane trajectories 123 Author s personal copy Numerical analyses and experimental validations of coexisting multiple attractors2363 Fig 3 For six values of k projections of various attractors on the x1 x3 plane a coexisting point attractors k 0 52 b period 3 attractor k 0 75 c double scroll chaotic attractor k 0 8 d coexisting single scroll chaotic attractors k 0 95 e coexisting period 4 attractors k 0 97 f coexisting period 1 attractors k 1 1 x1 x3 x1 x3 a x1 x3 x3 x1 c x1 x3 x1 x3 e b d f 4 1 Coexistence of single scroll chaotic attractors and periodic ones The coexisting multiple attractors behavior of a pair ofsingle scrollchaoticattractorsandapairofperiod 3 ones in the neighborhoods of k 0 95 is displayed in Fig 4 where the red blue black and green orbits correspond to the state initial values 0 0 01 0 0 0 01 0 1 0 1 and 1 0 1 respectively Four kinds of bifurcation diagrams in the range of 0 94 k 0 96 are simultaneously drawn in Fig 4a and the coexistence of a pair of single scroll chaotic attractors and a pair of period 3 ones is exhibited in Fig 4b For the symmetrical state initial values 0 0 01 0 and 0 0 01 0 three fi nite time Lyapunov exponents of 1 are 0 0659 0 0003 and 0 3538 respectively but for the symmetrical state initial values 1 0 1 and 1 0 1 three fi nite time Lyapunov exponents of 1 are 0 0000 0 0124 and 0 2730 respectively 4 2 Coexistence of periodic attractors with two periodicities Thecoexistingmultipleattractors behaviorofperiodic attractors with two periodicities in the neighborhoods ofk 0 9isillustratedinFig 5 wheretheredandblue orbits correspond to the state initial values 0 0 01 0 and 0 0 01 0 respectively whereas the black and green orbits correspond to the state initial values 0 0 1 0 and 0 0 1 0 respectively Four kinds of bifurcation diagrams in the range of 0 89 k 0 91 are together plotted in Fig 5a and the coexistence 123 Author s personal copy 2364B Bao et al Fig 4 Coexisting multiple attractors behavior of single scroll chaotic attractors and period 3 attractors near k 0 95 a bifurcation diagrams in the range of 0 94 k 0 96 b phase plane trajectories of a pair of single scroll chaotic attractors and a pair of period 3 attractors k x1 0 0 01 0 0 0 01 0 1 0 1 1 0 1 x1 x3 0 0 01 0 0 0 01 0 1 0 1 1 0 1 a b Fig 5 Coexisting multiple attractors behavior of periodic attractors with two periodicities near k 0 9 a bifurcation diagrams in the range of 0 89 k 0 91 b phase plane trajectories of a pair of period 2 attractors and a pair of period 3 ones k x1 0 0 01 0 0 0 01 0 0 0 1 0 0 0 1 0 x1 x3 0 0 01 0 0 0 01 0 0 0 1 0 0 0 1 0 a b Fig 6 Coexisting multiple attractors behavior of a double scroll chaotic attractor and period 1 ones near k 0 67 a bifurcation diagrams in the range of 0 66 k 0 68 b phase plane trajectories of a double scroll chaotic attractor and a pair of period 1 ones k x1 0 1 0 0 0 1 0 0 1 0 0 x1 x3 0 1 0 0 0 1 0 0 1 0 0 a b of a pair of period 2 attractors and a pair of period 3 ones is demonstrated in Fig 5b For the symmet rical state initial values 0 0 01 0 and 0 0 01 0 threefi nite timeLyapunovexponentsof 1 are0 0003 0 0050 and 0 3167 respectively whileforthesym metrical state initial values 0 0 1 0 and 0 0 1 0 threefi nite timeLyapunovexponentsof 1 are0 0001 0 1106 and 0 1598 respectively 4 3 Coexistence of a chaotic attractor and periodic ones Thecoexistingmultipleattractors behaviorofadouble scroll chaotic attractor and a pair of period 1 ones in the neighborhoods of k 0 67 is displayed in Fig 6 where the red blue and black orbits correspond to the state initial values 0 1 0 0 0 1 0 0 and 1 0 0 respectively Three kinds of bifurcation dia grams in the range of 0 66 k 0 68 are meanwhile depicted in Fig 6a and the coexistence of a double scroll chaotic attractor and a pair of period 1 ones is depicted in Fig 6b For the symmetrical state initial values 0 1 0 0 and 0 1 0 0 three fi nite time Lyapunov exponents of 1 are 0 0001 0 0412 and 0 5603 respectively while for the state initial values 1 0 0 three fi nite time Lyapunov exponents of 1 are 0 0443 0 0001 and 0 6661 respectively The coexisting multiple attractors behaviors with different attractor types different topology structures 123 Author s personal copy Numerical analyses and experimental validations of coexisting multiple attractors2365 or different periodicities in the HNN based system are revealed in Figs 4 5 and 6 which just embody the existence of complex dynamical characteristics with multistability 5 Hardware circuit developments and measurements Hardwarecircuitrealizationofmathematicalmodelsis a major subject from the point of practical engineering applications 10 13 26 29 44 45 47 Generally con tinuousnonlinearsystemscanbeimplementedbyusing commercially available electronic components inte grated IC circuits or FPGAs 7 22 29 37 47 48 By using already existing analog electronic components the hardware circuit of the HNN based nonlinear sys tem can be designed and fabricated upon which the generating phase plane trajectories can be observed by experimental instruments to validate numerical simu lations 5 1 Neuron activation function unit circuit The nonlinear term of the HNN based system is achieved by the neuron activation function which is a hyperbolic tangent function Therefore the key of hardware implementation of the HNN based system is to develop the circuit realization of a unit circuit of the hyperbolic tangent function Referring to the cir cuit design scheme proposed in 45 and utilizing the voltage currentexponentialcharacteristicofthecollec tor current of a bipolar NPN transistor a kind of circuit realization form of the neuron activation function unit is given in Fig 7 where the proportional constant cur rentsource I0isconstitutedbythetransistorpaircircuit in the dashed box Assumethatviandvorepresenttheinputandoutput voltages of the hyperbolic tangent function unit circuit in Fig 7 According to the analysis result in 45 the input outputrelationshipofthisunitcircuitcanbewrit ten as v0 tanh RF 2RVT vi 9 where VTis the thermal voltage of the transistor and its value is about 26 mV at room temperature When the circuit parameters are set as R 10k RF 520 RF VEE T1 RT R Ui vi T2 T3T4 RT R R R Uo RW vo RCRC VCC I0 R Fig 7 Circuit realization of hyperbolic tangent function unit circuit RC 1k RT 2k RW 9 8k VCC 15V and VEE 15V the coeffi cient of viin 9 can be simplifi ed as 1 i e the hyperbolic tangent function in 1 canbeimplemented Additionally theresultsofthe proportional constant current source I0measured from PSpice circuitsimulationandhardwarecircuitare1 17 and 1 1mA respectively which are very close 5 2 Schematics and equations of experimental circuit The operations of addition subtraction and integra tion of the system 1 can be completed by operational amplifi er connected with resistors and or capacitors 37 and the nonlinear function can be accomplished by the neuron activation function unit circuit given in Fig 7 Thus themaincircuitschemeoftheHNN based nonlinear system implemented by a pure analog circuit can be synthesized in Fig 8 where the circuit module marked by tanh in the box is the neuron activation function unit circuit in Fig 7 In the above circuit of Fig 8 v1 v2 and v3stand for the state variables of the capacitor voltages of three integral circuit channels respectively and RC is the integratingtimeconstant Hence thecircuitstateequa tions of Fig 8 can be formulated as Cdv1 dt 1 Rv1 1 R1 tanh v1 1 R2 tanh v2 1 R3 tanh v3 Cdv2 dt 1 Rv2 1 R4 tanh v1 1 R5 tanh v3 Cdv3 dt 1 Rv3 1 R6 tanh v1 1 R7 tanh v2 1 R8 tanh v3 10 123 Author s personal copy 2366B Bao et al tanh U1 Cv1 U2 R R R va va R1 vb va vc U3 Cv2 U4 R R R vb vb va vc U5 Cv3 U6 R R R vc vc vb va vc R3 R2 R4 R5 Ta1 Ta2 Ta3 R6 R8 R7 tanh tanh Fig 8 MaincircuitschemeoftheHNN basednonlinearsystem Set RC 100 s When the resistance of the inte grator is selected as R 10 k its capacitance can be chosen as C 10 nF On the basis of the synaptic weight matrix given in 2 other input resistances in Fig 8