论文集 2018_Numerical and experimental confirmations of quasi-periodic behavior and chaotic bursting in third-order autonomous memristive oscillator

Chaos Solitons and Fractals 106 2018 161 170 Contents lists available at ScienceDirect Chaos Solitons and Fractals Nonlinear Science and Nonequilibrium and Complex Phenomena journal homepage Numerical and experimental confi rmations of quasi periodic behavior and chaotic bursting in third order autonomous memristive oscillator B C Bao P Y Wu H Bao Q Xu M Chen School of Information Science and Engineering Changzhou University Changzhou 213164 China a r t i c l e i n f o Article history Received 17 September 2017 Revised 2 November 2017 Accepted 15 November 2017 Keywords Memristive diode bridge Autonomous memristive oscillator Quasi periodic behavior Chaotic bursting 0 1 test a b s t r a c t This paper presents a novel third order autonomous memristive chaotic oscillator which is accomplished by parallelly coupling a simple memristive diode bridge emulator into a Sallen Key low pass fi lter LPF With the modeling of this oscillator stability analyses of the equilibrium point and numerical simulations of the phase plane orbit time domain sequence bifurcation diagram and fi nite time Lyapunov exponent spectrum are performed from which period quasi period chaos and quasi period to chaos route are found Particularly two types of dynamical phenomena of quasi periodic behavior and point cycle chaotic bursting that are further identifi ed by using 0 1 test are observed in such a third order autonomous memristive oscillator which have been rarely reported in the previous literatures Additionally hardware experiments are implemented and the quasi periodic behavior and point cycle chaotic bursting are well confi rmed 2017 Elsevier Ltd All rights reserved 1 Introduction The emergence of physical memristors promotes the rapid de velopment of new circuit network theory In the past several years many physically realizable memristor emulators have been pre sented for breadboard experiment investigations of memristor based chaotic oscillating circuits 1 9 These memristor emula tors can be classifi ed into two categories op amps and analog multipliers based equivalent circuits 1 5 and memristive diode bridge cascaded with L RC or LC components 6 11 However most of these memristive chaotic oscillating circuits are at least fourth order autonomous 1 4 6 9 only the memristive band pass fi lter chaotic circuit reported in 5 is third order autonomous and its memristor emulator is equivalently realized by op amps and analog multipliers Differing from the memristive circuit in 5 this paper presents a third order autonomous memristive chaotic oscillator consisting of a Sallen Key low pass fi lter LPF and a memristive diode bridge emulator which is extended from the memristive circuit that was proposed by 11 and has same elec tronic components but different circuit topologies Chaotic bursting dynamics a specifi c dynamical phenomenon has been frequently exhibited in various kinds of neuronal mod els 12 21 which manifests a type of communication activity in biological neurons and endocrine cells 12 Distinguishing from the above neuronal dynamics an interesting symmetric point cycle chaotic bursting 12 is obtained by numerical simulations and ex Corresponding author E mail address mervinbao B C Bao perimental measurements in such a third order autonomous mem ristive chaotic oscillator which resembles the result reported in 11 and has been encountered in fast slow dynamical systems 22 24 and memristor based chaotic circuits 9 25 26 However these memristor based chaotic circuits are at least third order non autonomous 25 26 or fourth order autonomous 9 which re sult in the complexity of the realization circuit topologies or alge braic system structures With simple circuit topology and algebraic structure the memristive chaotic oscillator presented in this paper is only third order autonomous but can exhibit the same chaotic bursting dynamics as demonstrated in 11 Moreover another interesting quasi periodic phenomenon can be easily observed in this simple memristive chaotic oscillator and it has not been reported in any previous literatures yet Usually the quasi periodic behavior appears in some special nonlinear dynam ical systems of periodically forced and or delayed oscillators 26 28 relay feedback systems and turbocharger model 29 30 gen eralized Nos Hoover oscillator and quasi periodic self oscillator 31 32 biochemical cells and systems 33 34 high dimensional coupled chaotic systems 35 37 and 4 DOF isotropic oscillators along with real and fi nancial interacting markets 38 39 Neverthe less it should be emphasized that the proposed memristive chaotic oscillator is low dimensional and autonomous which can be taken as a demo for mathematically numerically and experimen tally investigating of complex nonlinear dynamical phenomenon Because quasi periodic oscillations behave like chaotic oscilla tions and chaotic bursting motions look similar to intermittent multi periodic motions an extra and effective 0 1 test method https doi org 10 1016 j chaos 2017 11 025 0960 0779 2017 Elsevier Ltd All rights reserved 162 B C Bao et al Chaos Solitons and Fractals 106 2018 161 170 C1 V1 R R1 R2 C2 U R V2 W D1D4 D2D3 V1 i i0 L i a b Fig 1 Sallen Key LPF based memristive chaotic oscillator a third order au tonomous memristive circuit b memristive diode bridge cascaded with an induc tor 40 44 can be employed to further confi rm the revealed quasi periodic behavior and chaotic bursting in this paper After this introduction the paper is organized as follows In Section 2 a novel third order autonomous Sallen Key LPF based memristive chaotic oscillator is constructed and then mathemat ical modeling MATLAB numerical simulations and equilibrium stability analyses are performed In Section 3 complex bifurca tion behaviors are revealed by conventional dynamical methods from which two types of specifi c dynamical phenomena of quasi periodic behavior and point cycle chaotic bursting are observed and further identifi ed by using 0 1 test Furthermore hardware experiments are implemented and the quasi periodic behavior and point cycle chaotic bursting are confi rmed in Section 4 The con clusions are summarized in Section 5 2 Sallen Key LPF based memristive oscillator A third order autonomous memristive chaotic oscillator is pre sented as shown in Fig 1 a which is constructed by a Sallen Key LPF parallelly coupling a memristive diode bridge emulator unlike that the memristive emulator is parallelly linked with the capacitor C 1 in the memristive chaotic circuit reported in 11 2 1 Mathematical modeling The emulator of memristive diode bridge cascaded with an in ductor is displayed in Fig 1 b Denote that V 1 and i are the input voltage and current of the emulator and i 0 is the current of the in ductor L Referring to 11 the voltage and current relation of the memristive diode bridge emulator is described as i W i 0 V 1 v i 0 2 I S tanh V 1 d i 0 d t 1 L ln 2 I S cosh V 1 1 L ln i 0 2 I S 1 where 1 2 nV T I S n and V T are the three diode model param eters and stand for the reverse saturation current emission coeffi cient and thermal voltage respectively For the sake of simplifying the mathematical model three vari ables of node voltage V 1 node voltage V 2 and inductor current i 0 are considered According to the fundamental theory of circuit the circuit equations of the third order autonomous memristive chaotic oscillator in Fig 1 can be formulated as d i 0 d t 1 L ln 2 I S cosh V 1 1 L ln i 0 2 I S d V 1 d t k C 1 2 C 2 V 1 R 1 C 1 1 k C 2 V 2 R i 0 2 I S tanh V 1 C 2 d V 2 d t k V 1 V 2 R C 1 2 where k R 2 R 1 1 Table 1 Circuit parameters of the third order autonomous memristive chaotic os cillator Parameters Signifi cations Values C 1 C 2 Capacitance 47 nF L Inductance 10 mH R Resistance 200 R 1 Resistance 3 k R 2 Resistance 6 6 k I S Reverse saturation current of 1N4148 diode 5 84 nA n Emission coeffi cient of 1N4148 diode 1 94 V T Thermal voltage of 1N4148 diode 25 mV To facilitate quantitative analysis the variables and circuit pa rameters of 2 are normalized in a dimensionless form as x R i 0 y V 1 z V 2 t R C 1 C 1 C 2 a R 2 C 1 L b 2 R I S k R 2 R 1 1 3 A set of ordinary differential equations are thereby deduced from 2 as x a ln b cosh y a ln x b y k 2 y 1 1 k z x b tanh y z ky z 4 In Fig 1 the circuit parameters of linear elements and nonlin ear 1N4148 diodes are selected as listed in Table 1 Therefore the typical normalized parameters of 4 are calculated by 3 as a 0 188 b 2 4082 10 5 and k 3 2 5 2 2 Quasi periodic and chaotic bursting oscillations The system parameters and the initial conditions are main tained as b 2 4082 10 5 k 3 2 and 0 0 001 0 For two dif ferent values of the inductor parameter as L 10 mH and 21 mH i e a 0 188 and 0 0895 the numerically simulated quasi periodic and chaotic bursting oscillations for the third order autonomous memristive chaotic oscillator are demonstrated by the phase plane orbits in the y z and y x planes time domain sequence of the variable y and three fi nite time Lyapunov exponent spectra calcu lated by Wolf s method 45 as shown in Figs 2 and 3 respec tively where MATLAB ODE23s algorithm with time step 0 02 s are used Initially it can be observed from Fig 2 that an infusive quasi periodic behavior is triggered in the third order autonomous mem ristive chaotic oscillator Fig 2 a and b display two projections of a quasi periodic limit cycle on two phase planes Fig 2 c il lustrates an intermittent multi periodic sequence of the variable y and Fig 2 d gives three fi nite time Lyapunov exponent spectra against time Glanced from Fig 2 a and b the moving trajecto ries on two phase planes are something like the chaotic motions However in Fig 2 d the values of the Lyapunov exponent spec tra at 20 ks are LE 1 0 0 LE 2 0 0 and LE 3 1 9855 respec tively which indicate that the quasi periodic motion occurs in such a third order autonomous memristive chaotic oscillator due to the appearance of two zero exponents and one negative exponent Afterwards it can be found from Fig 3 that a symmetric point cycle chaotic bursting phenomenon 12 similar to that re ported in 11 is emerged from the third order autonomous mem ristive chaotic oscillator whose attractor topology in the three dimensional phase space has apple shaped chaotic attractor struc ture Fig 3 a and b show two projections of a chaotic attrac tor on two phase planes with fractal structure Fig 3 c exhibits an aperiodic sequence with bursting oscillation and Fig 3 d pro vides three fi nite time Lyapunov exponent spectra against time The values of the Lyapunov exponent spectra at 20 ks are B C Bao et al Chaos Solitons and Fractals 106 2018 161 170 163 z x a b y Lyapunov Exponents 10 3 c d Fig 2 Numerically simulated quasi periodic oscillation at L 10 mH where a 0 188 b 2 4082 10 5 and k 3 2 a phase plane orbit in the y z plane b phase plane orbit in the y x plane c time domain sequence of y d three fi nite time Lyapunov exponent spectra against time z x a b y Lyapunov Exponents 10 3 c d Fig 3 Numerically simulated chaotic bursting oscillation at L 21 mH where a 0 0895 b 2 4082 10 5 and k 3 2 a phase plane orbit in the y z plane b phase plane orbit in the y x plane c time domain sequence of y d three fi nite time Lyapunov exponent spectra against time 164 B C Bao et al Chaos Solitons and Fractals 106 2018 161 170 Fig 4 Dynamical behaviors with respect to the inductor parameter L where a 0 94 0 0723 b 2 4082 10 5 k 3 2 and 0 0 001 0 a bifurcation diagram of the local maxima denoted by Y of the variable y b three fi nite time Lyapunov exponents with the time end 20 ks Fig 5 Numerically simulated phase plane orbits in the y z plane and time domain sequences of the variable y for different values of the inductor parameter L where a1 c1 are phase plane orbits and a2 c2 are time domain sequences a period at L 4 mH a 0 47 b quasi period at L 15 mH a 0 1253 c chaos at L 26 mH a 0 0723 B C Bao et al Chaos Solitons and Fractals 106 2018 161 170 165 Fig 6 The mean square displacement M versus n the left sides and p q dynamics the right sides using 0 1 test for different values of L or a a period at L 4 mH a 0 47 b quasi period at L 10 mH a 0 188 c quasi period at L 15 mH a 0 1253 d chaos at L 21 mH a 0 0895 e chaos at L 26 mH a 0 0723 Here N 50 0 0 and n 1 2 500 166 B C Bao et al Chaos Solitons and Fractals 106 2018 161 170 15 V C1 C2 R1 R2 R D1 L Sallen Key LPF Memristive diode bridge GND 15 V R D4 D3 D2 U V1 V2 Fig 7 Photograph of the experimental prototype for the third order autonomous memristive chaotic oscillator the left is a global graph of the digital oscilloscope connecting with the circuit breadboard and the right is an enlarged view of the circuit breadboard Table 2 Chaos indicator using 0 1 test for fi ve different values of L or a Values of L or a Dynamical behaviors c K 4 mH 0 47 Period 0 8701 0 00357 10 mH 0 188 Quasi period 0 8720 0 00715 15 mH 0 1253 Quasi period 0 9447 0 00860 21 mH 0 0895 chaos 2 1227 0 94866 26 mH 0 0723 chaos 2 2045 0 99594 LE 1 0 0099 LE 2 0 0017 and LE 3 44 8478 respectively fur ther signifying the existence of the chaotic behavior It should be remarked that the point cycle chaotic bursting oscillation is sym metric bipolar whereas the neuronal bursting oscillations are gen erally asymmetric unipolar 12 21 2 3 Stability analysis Obviously only one equilibrium point can be found in the nor malized system modeled by 4 which is expressed as P 0 0 0 6 For the equilibrium point P the Jacobian matrix is yielded as J a b 0 0 0 k b 2 1 k 1 0 k 1 7 The characteristic polynomial is thereby formulated as P a b 2 k b 3 b 1 0 8 Thus for the normalized parameters given by 5 the eigenval ues are given by solving 8 as 1 2 0 10 0 0 j0 9950 3 7804 58 9 For the determined parameters of b and k the complex con jugate roots of 1 2 remain unchanged Consequently P is always an unstable saddle focus in the passive region of the parameter a from which periodic and chaotic behaviors may be emerged in the third order autonomous memristive chaotic oscillator 3 Complex dynamics in the memristive oscillator To more expediently reveal the dynamical behaviors of the third order autonomous memristive chaotic oscillator the induc tor parameter L varied from 2 mH to 26 mH is adopted as an ad justing parameter which means that the system parameter a is changed from 0 94 to 0 0723 Meanwhile MATLAB ODE23s algo rithm is used for the dynamical analyses and Wolf s method 45 is utilized to calculate the fi nite time Lyapunov exponents with the time end 20 ks 3 1 Bifurcation behaviors With the variation of the inductor parameter L the bifurca tion diagram of the local maxima denoted by Y of the state vari able y and the three fi nite time Lyapunov exponents are drawn in Fig 4 a and b respectively from which periodic quasi periodic and chaotic behaviors along with bifurcation route from quasi period to chaos are discovered When the value of L is gradually adjusted in the region 2 00 mH 5 50 mH the third order au tonomous memristive chaotic oscillator always operates in peri odic oscillation state When the value of L is located in the region 5 60 mH 15 40 mH the memristive chaotic oscillator is mainly in quasi periodic oscillation state While when the value of L is settled down in the region 15 50 mH 26 mH the memristive chaotic oscillator is nearly in chaotic oscillation state It is noticed that several narrow periodic windows occur in the quasi periodic and chaotic regions and the point cycle chaotic bursting phenom ena mainly appear in the region 20 mH 26 mH For three specifi ed values of the inductor parameter L the phase plane orbits and time domain sequences using for demon strating different dynamical behaviors are given as shown in Fig 5 In more detail Fig 5 a displays a periodic oscilla tion at L 4 mH a 0 47 with the fi nite time Lyapunov expo nents LE 1 0 0 LE 2 0 0331 and LE 3 0 3572 Fig 5 b illus trates another quasi periodic oscillation at L 15 mH a 0 1253 with the fi nite time Lyapunov exponents LE 1 0 0 LE 2 0 0 and LE 3 11 8067 and Fig 5 c exhibits another chaotic behavior at B C Bao et al Chaos Solitons and Fractals 106 2018 161 170 167 Fig 8 Experimentally measured phase plane orbits in the V 1 t V 2 t plane and time domain sequences of the node voltage V 1 t for different values of the inductor parameter L where a1 e1 are phase plane orbits and a2 e2 are time domain sequences a periodic oscillation at L 4 mH b quasi periodic behavior at L 10 mH c quasi periodic behavior at L 15 mH d chaotic bursting at L 21 mH e chaotic bursting at L 26 mH L 26 mH a 0 0723 with the fi nite time Lyapunov exponents LE 1 0 0166 LE 2 0 0 and LE 3 53 6222 3 2 The 0 1 test results According to the inductor parameter L associated dynamical behaviors and phase plane orbits numerically simulated in Figs 2 3 and 5 respectively it can be revealed that the third order autonomous memristive chaotic oscillator shows the transitions from periodic limit cycles to quasi periodic limit cycles and then to chaotic attractors Especially identifying the quasi periodic and chaotic bursting motions is still not easy due to that quasi period behaves like chaos and chaotic bursting looks similar to intermit tent multi period Herein the 0 1 test for chaos is utilized to fur ther confi rm the quasi periodic and chaotic dynamics shown in Figs 2 3 and 5 40 44 The 0 1 test can be executed by analyzing any observable vari able associated with the system dynamical response 40 41 For an arbitrary constant c in the region of 0 two additional vari 168 B C Bao et al Chaos Solitons and Fractals 106 2018 161 170 Fig 8 Continued ables p and q can be defi ned as p n n j 1 j cos jc q n n j 1 j sin jc 10 where n j 1 N These new variables p and q change over time as a consequence of the evolution of the observable variable which just refl ect the information of the system dynamical re sponse When the system dynamics is regular two additional vari ables p and q show a bounded evolution