2017年论文 2017_Sallen–Key low-pass filter-based inductor-free simplified Chua’s circuit

Sallen Key low pass fi lter based inductor free simplifi ed Chua s circuit Han Bao Pingye Wu Bo Cheng Bao Mo Chen Huagan Wu School of Information Science and Engineering Changzhou University Changzhou 213164 Jiangsu People s Republic of China E mail mervinbao Published in The Journal of Engineering Received on 21st July 2017 Accepted on 24th July 2017 Abstract This present study reports a new type of inductor free simplifi ed Chua s circuit only using two op amps based realisation which is implemented by cascading a Sallen Key low pass fi lter with a simplifi ed Chua s diode With mathematical model based numerical simulations and hardware circuit based experimental measurements it is demonstrated that the reported circuit exhibits the similar dynamical behaviours of the classical Chua s circuit such as point attractor limit cycles period doubling route chaotic spiral attractors double scroll chaotic attractor boundary crisis coexisting bifurcation modes and so on Furthermore the proposed circuit enriches the realisation forms of Chua s circuit family 1Introduction ThewellknownChua scircuitisthe fi rstautonomous chaotic circuit 1 2 which has been regarded as a paradigm in the studying of chaos 3 However the classical Chua s circuit has at least an inductor which makes practical fabrication com plicated and less robust For seeking a simple inductor free Chua s circuit a new type of inductor free simplifi ed Chua s circuit only using two op amps based realisation is presented which is achieved by cascading a Sallen Key low pass fi lter LPF 4 with a simplifi ed Chua s diode 5 Different from the inductor free simplifi edChua scircuitreportedin 6 an active LPF is utilised to substitute the Wien bridge oscillator or active band PF Nevertheless the newly constructed chaotic circuit has the similar dynamical behaviours of the classical Chua s circuit and expands the family of Chua s circuit and Chua s equation 2 Sallen Key LPF based inductor free simplifi ed Chua s circuit The new type of inductor free simplifi ed Chua s circuit is drawn in Fig 1 where the right circuit of the port 11 is a simplifi ed real isation of Chua s diode 6 but the left circuit of the port 22 is a Sallen Key LPF For the voltage vNacross and the current iNthrough the simplifi ed Chua s diode RN the relation between vNand iNis described as 6 iN h vN GbvN 0 5 Ga Gb vN Bp vN Bp 1 whereGa Gb andBparethreerealconstants Forthe saturation voltage Esatof the op amp and the condition Rb Rc there yield Ga 1 Ra Gb 1 Rb Bp Ra Ra Rb Esat 2 With the three state variables of v1 v2 and v3given in Fig 1 the state equations of the newly simplifi ed Chua s circuit are modelled as C1 dv1 dt 1 R v1 kv2 v3 h v1 C2 dv2 dt 1 R0 k 1 v2 v3 C3 dv3 dt 1 R v1 k R 2k 1 R0 v2 1 R 2 R0 v3 3 where k Rf Ri 1 By opening the three capacitors in Fig 1 the DC equilibrium points of the Chua s circuit can be easily obtained which is expressed as S0 0 0 0 S R0 R I0 R0I0 k 1 R0I0 4 where I0 Ga Gb Bp R0 R Gb 1 The Jacobian matrix at the equilibrium point S v1 v2 v3 is derived as J 1 RC1 h v1 C1 k RC1 1 RC1 0 k 1 R0C2 1 R0C2 1 RC3 k RC3 2k 1 R0C3 1 RC3 2 R0C3 5 where h v1 Gb 0 5 Ga Gb sign v1 Bp sign v1 Bp Fig 1 Sallen Key LPF based inductor free simplifi ed Chua s circuit J Eng 2017 doi 10 1049 joe 2017 0311 This is an open access article published by the IET under the Creative Commons Attribution License http creativecommons org licenses by 3 0 For the Chua s circuit shown in Fig 1 the coupling resis tanceisuniquelyadjustableanditstypicalvalueis R 1 5 k other circuit parameters are determined as Ra 1 4 k Rb Rc 10 k R0 200 Ri 3 k Rf 6 6 k C1 10 nF and C2 C3 220 nF Withthetypicalcircuitparameters thetwonon zero equilibrium points can be calculated as S 1 8889 0 2222 0 4889 and the eigenvalues for the three equilibrium points are solved from 5 as S0 l1 14483 97 l2 3 0 4103 j2 2195 S l1 2 0 1579 j2 2627 l3 78309 15 6 which indicate that S0is a saddle focus of index 1 and S is a pair of saddle foci of index 2 analogous to those of the classical Chua s circuit 3Numerical simulations and hardware experiments Take the coupling resistance R as a bifurcation parameter When R is gradually changed from 1 1 to 2 2 k and the initial conditions of the red and blue trajectories are respectively fi xed as v1 0 v2 0 v3 0 1 mV 0 V 0 V the bifurcation diagrams of the state variable v2are displayed in the upper part of Fig 2 Correspondingly the fi rst two Lyapunov exponents for 1 mV 0 V 0 V are calculated in the lower part of Fig 2 The results in Fig 2 signify that the circuit in Fig 1 emulates the dynamical beha viours of the classical Chua s circuit such as point attractor limit cycle period doubling route chaotic spiral attractor double scroll chaotic attractor boundary crisis coexisting bifurcation mode and so on With the circuit schematic given in Fig 1 a hardware circuit on a breadboard is made where three monolithic ceramic capacitors seven precision metal fi lm resistors one precision potentiometer and two op amps AD711KN with 15 V voltage sources are chosen For different values of R the phase portraits of several dif ferent attractors in the v1 t v2 t plane are numerically simulated and experimentally captured as shown in Fig 3 Observed from Fig 3 point attractor coexisting right left limit cycles coexisting right left chaotic spiral attractors and double scroll chaotic attract or are found which can effectively exhibit the dynamical beha viours of the newly proposed Chua s circuit 4Conclusions By cascading Sallen Key LPF with simplifi ed Chua s diode a new inductor free simplifi ed Chua s circuit is presented in this paper which can exhibit the similar dynamical behaviours of the classical Chua s circuit Especially Sallen Key LPF based simplifi ed Chua s circuit with simply physical realisation is inductor free and robust suitable for theoretical and experimental demonstrations of chaos and useful for chaos based engineering applications 5Acknowledgments This work was supported by the National Natural Science FoundationofChina grants51607013 61601062 and 51277017 and the Natural Science Foundations of Jiangsu Province China grant BK20160282 6References 1 Matsumoto T A chaotic attractor from Chua s circuit IEEE Trans Circuits Syst 1984 CAS 31 12 pp 1055 1058 2 Kennedy M P Robust op amp realization of Chua s circuit Frequenz 1992 46 3 4 pp 66 80 Fig 2 Dynamical behaviours with the variation of the resistance R Fig 3 Attractors using R 1 1 1 3 1 35 1 4 1 5 k respectively a1 e1 Numerical simulated results in the v1 t v2 t plane a2 e2 Experimental captured results in the v1 t v2 t plane This is an open access article published by the IET under the Creative Commons Attribution License http creativecommons org licenses by 3 0