2018年论文 2018_Coexistence of multiple bifurcation modes in memristive diode bridge based canonical Chua s circuit

Full Terms memristive diode bridge multiple bifurcation modes coexisting attractors hardware experiment 1 Introduction Because of nature non linearity and plasticity of memristors Strukov Snider Stewart Chen Xu Lin Kennedy 1992 Premlet 2011 various kinds of memristor based Chua s circuits and their non linear phenomena have received great attention and large amounts of research achievements have been reported in the past few years Bao et al 2017 Bao Jiang Wu Bao Xu Bao Chen et al 2015 Chen Yu Yu Li Fitch Yu Iu Wen Zeng Xu Lin Bao Xu Wang Bao Chen Chen et al 2015 2014 Fitch et al 2012 Wen et al 2012 Xu et al 2016 2016 Based on these memristive Chua s chaotic circuits new properties of line plane equilibrium Bao et al 2017 2015 2016 Fitch et al 2012 Wen et al 2012 chaos and transient chaos Bao et al 2017 2015 hyperchaos Fitch et al 2012 coexisting multiple attractors Xu et al 2016 coexisting infi nitely many attractors Bao et al 2017 2016 and self excited hidden oscillations Chen et al 2015 have been discovered and analysed in detail In special generalised memristor emulators consisting of memristive diode bridge cascaded with fi rst order parallel resistor and capacitor RC or second order parallel inductor and capacitor LC fi lters Bao Yu Hu Wu Bao Liu Xu Kengne Tabekoueng Temba Njitacke Kengne Fotsin Negou Wu et al 2016 Xu et al 2016 Due to the introduction of memristor the dynamical behaviours of a memristor based chaotic circuit are closely associated with memristor initial value Bao et al 2017 2016 resulting in the emergence of multistability Kengne et al 2015 Njitacke et al 2016 Xu et al 2016 or extreme multistability Bao et al 2017 2016 Multistability or extreme multistability i e coexisting phe nomenon of multiple attractors or infi nitely many attractors is an interesting and striking non linear phenomenon which is an initial condition dependent dynamical behaviour and usually encounters in some non linear dynamical systems Hens Banerjee Feudel Hens Dana Jaros Perlikowski Li Hu Sprott Li Pehlivan Sprott Li Ngouonkadi Fotsin Fotso Tamba Pham Volos Kapitaniak Jafari Yuan Wang Morfu Nofi ele Pisarchik v v i0 2IS tanh v 1 L0 di0 dt 1 ln 2IScosh v 1 ln i0 2IS R0i0 2 where 1 2nVT three diode model parameters of Is n and VTdenote the reverse saturation current emission coeffi cient and thermal voltage respectively Note that equation 2 is a simplifi ed expres sion which is derived from the original model in Wu et al 2015 If four diodes 1N4148 are used in Figure 1 b the diode model parameters are Is 5 84 nA n 1 94 and VT 25 mV respectively 2 2 Mathematical model of the memristive circuit The modifi ed memristive canonical Chua s circuit shown in Figure 1 a is composed of capacitor C1 capacitor C2 inductor L1 negative resistor RNand memristor GM With the four state variables of v1 v2 i1and i0 the circuit state equations in Figure 1 a can be described as C1 dv1 dt i1 i0 2IS tanh v1 C2 dv2 dt v2 RN i1 L1 di1 dt v2 v1 L0 di0 dt 1 ln 2IScosh v1 1 ln i0 2IS R0i0 3 By introducing four new variables and scaling the circuit parameters in a dimensionless form as x v1 y v2 z RNi1 w RNi0 t RNC2 a C2 C1 b R2 NC2 L1 c 2 RNIS d1 R2 NC2 L0 d2 R0RNC2 L0 4 The normalised system equations can be rewritten as x az a w c tanh x y y z z b y x w d1ln ccoshx d1ln w c d2w 5 Figure 1 Memristive chaotic circuit and memristor emulator a Modifi ed memristive canonical Chua s circuit b Memristive diode bridge with series RL fi lter INTERNATIONAL JOURNAL OF ELECTRONICS1161 Correspondingly for the circuit parameters listed in Table 1 the normalised system parameters are calculated from equation 4 as a 4 7 b 2 1733 c 8 1880 10 4 d1 21 7328 d2 1 5980 6 2 3 Stabilities of equilibrium points The equilibrium points of equation 5 can be yielded by solving the following equations 0 az a w c tanhx 0 y z 0 b y x 0 d1ln ccoshx d1ln w c d2w 7 Obviously the equilibrium points of equation 5 can be determined as a zero equilibrium point P0 0 0 0 0 and two non zero equilibrium points P1 x1 y1 z1 w1 and P2 x2 y2 z2 w2 where P1and P2are calculated by x1 y1 z1 1 w1 1coth 1 c 8a and x2 y2 z2 2 w2 2coth 2 c 8b The values of 1and 2are two roots of the following transcendental equation d1ln ccosh d1ln coth d2 coth c 0 9 Certainly the values of 1and 2 are fi xed through graphic analytic method Chen et al 2015 2014 Xu et al 2016 The Jacobian matrix at the equilibrium point P x y z w is derived from equation 5 as J a w c sech2 x 0a atanh x 01 10 bb00 d1tanh x00 d1 w c d2 2 6 6 4 3 7 7 5 10 Therefore the characteristic polynomial equation is given as P 4 3 3 2 2 1 0 0 11 where 3 d2 1 d1 tanh 2a sinh2 2 b d2 ab ad1 d1 tanh 2a d2 1 sinh2 1 b d2 ad2 a ad1 1 a bd1 tanh 2a b d2 sinh2 0 ab d1 d2 abd1 tanh 2abd2 sinh2 12 Table 1 Circuit parameters of the modifi ed memristive canonical Chua s circuit Parameters Signifi cationsValues C1Capacitance1 nF C2Capacitance4 7 nF L0Inductance10 mH L1Inductance100 mH RNResistance6 8 k R0Resistance0 5 k 1162B BAO ET AL With the normalised system parameters determined by equation 6 the values of 1and 2can be solved from equation 9 as 1 2 11 009 and the two non zero equilibrium points are thereby deduced as P1 2 11 009 11 009 11 009 11 008 Therefore the eigenvalues for the three equilibrium points can be obtained by solving the Jacobian matrix 10 as P0 1 0 8333 2 3 0 0814 j3 4988 4 26544 P 1 2 0 3298 j1 2474 3 4 1 6158 j10 4315 Obviously P0is an unstable saddle focus having one positive real root two complex conjugate roots with positive real parts and one negative real root whereas P are unstable saddle foci having two complex conjugate roots with positive real parts and another two complex conjugate roots with negative real parts 3 Numerical and experimental results 3 1 Numerical simulations For the normalised system parameters in equation 6 the given model parameters of diode 1N4148 and the initial values 0 01 0 0 0 the numerically simulated phase portraits of typical chaotic attractor in four diff erent planes are drawn in Figure 2 which demonstrate that the typical chaotic attractor is double scroll with complicated topological structure To investigate the rich variety of bifurcation behaviours that can be observed in the modifi ed memristive canonical Chua s circuit we solve numerically the model 5 using Runge Kutta ODE45 algorithm Taking the control parameter d2 i e the resistance R0 as a bifurcation parameter within the interval range from 0 5 to 8 the two superposed bifurcation diagrams of the state variable x are depicted in Figure 3 a where the orbits coloured in blue and red start from the initial values of 0 01 0 0 0 and 0 01 0 0 0 respectively Correspondingly the three Lyapunov exponents for the initial values of 0 01 0 0 0 are calculated by Wolf s method Wolf Swift Figure 2 Numerically simulated phase portraits of typical chaotic attractor a Phase portrait in the x y plane b Phase portrait in the x z plane c Phase portrait in the y z plane d Phase portrait in the y w plane Figure 3 Bifurcation behaviours with respect to d2 a Bifurcation diagrams of x b Lyapunov exponent spectra INTERNATIONAL JOURNAL OF ELECTRONICS1163 Swinney Dawson Grebogi Yorke Kan Kengne Negou Chen et al 2017 Kennedy 1992 Premlet 2011 In addition a 4 channel digital oscilloscope in the XY mode is used to capture the experimental phase portraits Corresponding to the numerically simulated chaotic attractor shown in Figure 2 the experimentally captured phase portraits in four diff erent planes are given in Figure 5 In the same way for diff erent values of d2 the experimentally captured phase portraits in the v1 v2plane are obtained in Figure 6 where the desired values of the system parameter d2are obtained by adjusting the resistor R0 whose resistances can be determined by the relation of R0 d2L0 RNC2 Comparing Figure 2 with Figure 5 and Figure 4 with Figure 6 the simulated and captured results are in basic agreement when ignoring the errors caused by the equiva lent series resistances of two manually winding inductors and other parasitic parameters Figure 4 Numerically simulated phase portraits under diff erent values of d2 a Period 3 limit cycle at d2 4 1 b Double scroll chaotic attractor at d2 4 3 c Coexisting right left period 2 limit cycle at d2 5 9 d Coexisting right left period 3 limit cycle at d2 6 9 Figure 5 Experimentally captured phase portraits of the typical chaotic attractor a Phase portrait in the v1 v2plane b Phase portrait in the v1 i1plane c Phase portrait in the v2 i1plane d Phase portrait in the v2 i0plane 1164B BAO ET AL Note that AC coupling modes are used for the voltage and current probes during hardware experiments and the coexisting attractors with diff erent colours in Figures 6 c and d are post processed 4 Coexistence of multiple bifurcation modes Taking a very sharp observation of the bifurcation diagrams of Figure 3 a crisis scenario is identifi ed in a narrow region of 6 24 d2 6 30 which results in the sudden change of the orbits This particular bifurcation behaviour indicates that multiple bifurcation modes occur in the modifi ed memristive canonical Chua s circuit leading to the emergence of multiple attractors To highlight the coexisting behaviours of multiple bifurcation modes six sets of diff erent initial values are specifi ed as 1 0 0 0 0 4 0 0 0 and 0 01 0 0 0 respectively When the system parameter d2is increased in the region of 6 2 6 4 six kinds of bifurcation diagrams of the state variable x accompanied with the corresponding Lyapunov exponents under diff erent initial values are displayed in Figure 7 where the coexistence of multiple bifurcation modes can be intuitively observed Note that in Figure 7 a the orbits marked with red blue cyan magenta green and brown colours correspond to those started from the initial values 1 0 0 0 1 0 0 0 0 4 0 0 0 0 4 0 0 0 0 01 0 0 0 and 0 01 0 0 0 respectively For two sets of the initial values 1 0 0 0 two period doubling bifurcation routes are presented With respect to the increase of the system parameter d2 the moving trajectories start from period 3 behaviours and then transform into period 6 behaviours via period doubling bifurcation routes Correspondingly two zero maximum Lyapunov exponents are always main tained for these two sets of the initial values However for other four sets of the initial values 0 4 0 0 0 and 0 01 0 0 0 the moving trajectories start from period 3 behaviours with zero maximum Lyapunov exponents and then suddenly jump into chaotic behaviours with positive maximum Lyapunov exponents at d2 6 23 As d2further increases the chaotic spiral attractors follow four reverse period doubling bifurcation routes accompanied with the occurrence of periodic behaviours with zero maximum Lyapunov exponents and fi nally mutate into period 6 behaviours at d2 6 33 and d2 6 30 respectively It should be stressed that the two reverse period doubling bifurcation routes for the initial values 0 4 0 0 0 are complete whereas those for 0 01 0 0 0 are incomplete Consequently under diff erent initial conditions six types of bifurcation routes are exhibited and four sets of topologically diff erent and disconnected attractors are clearly observed within the specifi ed parameter region For the above diff erent initial values when d2 6 25 is considered a kind of coexisting multiple attractors is together shown in Figure 8 a1 where the Lyapunov exponents for the right left period 3 limit cycles and right left chaotic spiral attractors are calculated as L1 0 0001 L2 0 0116 L3 2 9598 and L4 12 9270 as well as L1 0 0712 L2 0 0001 L3 3 0637 and L4 15 8351 respectively When d2 6 32 is selected another kind of coexisting multiple attractors is together shown in Figure 8 b1 where the Lyapunov Figure 6 Experimentally captured phase portraits under diff erent resistances of R0 a Period 3 limit cycle at R0 0 87 k b Double scroll chaotic attractor at R0 1 02 k c Coexisting right left period 2 limit cycle at R0 1 46 k d Coexisting right left period 3 limit cycle at R0 1 87 k INTERNATIONAL JOURNAL OF ELECTRONICS1165 exponents for the right left period 6 limit cycles and right left period 2 limit cycles are calculated as L1 0 0001 L2 0 0706 L3 2 9071 and L4 12 9040 along with L1 0 0001 L2 0 0167 L3 2 9863 and L4 15 0434 respectively In the experimental measurements the resistances of R0are adjusted as 1 67 and 1 83 k respectively When repeatedly powering on and off the hardware equipment of the modifi ed memristive canonical Chua s circuit two types of coexisting multiple attractors are displayed in Figures 8 a2 and b2 respectively It should be mentioned that the desired initial voltages and currents are diffi cultly determined in hardware experimental measurements so the initial values in the experimental results are randomly sensed by repeatedly powering on and off the hardware equipment As long as the sensed initial values locate within one of attraction basins of coexisting multiple attractors a type of attractor can be physically captured Figure 7 Coexisting behaviours of multiple bifurcation modes under six sets of diff erent initial values where a1 b1 and c1 are the bifurcation diagrams a2 b2 and c2 are the corresponding Lyapunov exponents a Coexisting behaviours under 1 0 0 0 b Coexisting behaviours under 0 4 0 0 0 c Coexisting behaviours under 0 01 0 0 0 1166B BAO ET AL 5 Conclusion This paper presents a modifi ed memristive canonical Chua s circuit and investigates its complex dynamical behaviours and the coexistence of multiple bifurcation modes therein With the circuit schematics of canonical Chua s circuit and memristive diode bridge emulator a normal ised system model of the proposed memristive circuit is built and thereby its equilibrium points and their stability analyses are performed Based on numerical simulations bifurcation beha viours related to the system parameter and initial values are investigated The research results verifi ed by hardware experiments demonstrate that six types of bifurcation modes are coexisted under six sets of diff erent initial values resulting in the coexistence of four sets of topologically diff erent and disconnected attractors The multiple bifurcation modes manifest that the pro posed memristive circuit is a multistable non linear circuit which is easy to be practically implemented with commercially available components and especially suitable for teaching aids and breadboard experiments of complex dynamical phenomenon Bertacchini Bilotta Pantano Tavernise 2012 To explore potential engineering applications of such a circuit proper control strategies should be applied to direct it to a desired stable state which could be an issue of our further research Acknowledgements This work was supported by the grants from the National Natural Science Foundations of China Grant Numbers 51777016 61601062 51607013 and 11602035 and the Natural Science Foundations of Jiangsu Province China Grant Number BK20160282 Figure 8 Coexistences of two types of topologically diff erent and disconnected multiple attractors under diff erent initial values where a1 and b1 are numerically simulated phase portraits in the x y plane a2 and b2 are experimentally captured phase portraits in the v1 t v2 t plane a Right left period 3 limit cycles and right left chaotic spiral attractors a Right left period 6 limit cycles and right left period 2 limit cycles INTERNATIONAL JOURNAL OF ELECTRONICS1167 Disclosure statement No potential confl ict of interest was reported by the authors Funding This work was supported by the National Natural Science Foundations of China Grant Numbers 11602035 51607013 51777016 61601062 and Natural Science Foundations of Jiangsu Province China Grant Number BK20160282 References Bao B C Jiang P Wu H G Hu F W 2015 Complex transient dynamics in periodically forced memristive Chua s circuit Nonlinear Dynamics 79 4 2333 2343 Bao B C Jiang T Wang G Y Jin P P Bao H Chen M 2017 Two memristor based Chua s hyperchaotic circuit with 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