2017年论文 2017_Emerging multi-double-scroll attractor from variable-boostable chaotic system excited by multi-level pulse

Emerging multi double scroll attractor from variable boostable chaotic system excited by multi level pulse Ning Wang Bo Cheng Bao Quan Xu Mo Chen Ping Ye 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 29th September 2017 Revised on 6th November 2017 Accepted on 14th November 2017 Abstract This study reports a simple approach to emerging multi double scroll attractor from a variable boostable chaotic system via multi level pulse excitation which is achieved by replacing a constant controller with a multi level pulse function To intuitively show the simple and feasible approach a four double scroll chaotic attractor coined in such a system with four level pulse excitation is taken as a paradigm 1Introduction In the past few decades generating multi scroll chaotic attractors from non linear dynamical system has always been a research topic of chaos based theories and applications due to higher topology complexity of multi scroll chaotic attractors than that with few scrolls 1 4 Most of these implementing approaches can be divided into two cases autonomous and non autonomous However theseworksachievedviaautonomousapproach often contain complex circuit implementations From the point view of simplifying circuit implementation and practical applica tion non autonomous approach has remarkable advantage due to the employment of signal generator 3 4 The signal generator can be easily realised by a ready made signal voltage current source A new regime of chaotic systems called variable boostable chaotic systems is explored in 5 whose compelling feature is that one of the variables has the freedom of offset boosting leading to the excellent convenience for signal transformation and transmission 5 6 In this paper a simple approach to generate multi double scroll attractor is introduced by replacing the constant controller in the variable boostable chaotic systems with a multi level pulse excitation The newly proposed method is simple and feasible for circuit implementation and practical application 2Constructing scheme of multi double scroll system According to the defi nition given in 5 a variable boostable chaotic system with the freedom for offset boosting the variable x is expressed as x yz y 1 z2 z x c yz 1 where the real constant c is a boosting controller of the variable x along the x axis Positive c boosts the variable x in the negative dir ection otherwise negative c boosts the variable x in the positive direction Note that the variation of c does not change the dynamics of system 1 5 With a pair of symmetry equilibrium points c 0 1 system 1 can generate a double scroll chaotic attractor whose scroll locations are determined by c along the x axis If the constant c is substituted with a periodic piecewise function the double scroll chaotic attractor will shift to different locations within different timeintervals therebyresultingintheemergenceof multi double scroll chaotic attractor As a result a non autonomous approach to generating multi double scroll attractor can be easily constructed A multi level pulse function f t is taken as the periodic piece wise function which can be described as 3 4 f t n 1 Ansgn sin 2pfnt 2 where Anand fnstand for the amplitude and frequency of the n th pulse excitation respectively Regarding n 2 as a paradigm then yields f t A1sgn sin 2pf1t A2sgn sin 2pf2t 3 When the parameters used in 3 are fi xed as A1 16 15 A2 8 15 f1 0 001 Hz and f2 0 002 Hz a four level periodic pulse function can be built By introducing 3 to substitute the constant boosting controller c in system 1 a novel variable boostable chaotic system excited by four level pulse is modelled as x yz y 1 z2 z x f t yz 4 Obviously system 4 has eight alternating current AC equilib rium points with the time evolutions which are calculated as xe ye ze 8 5 0 1 8 15 0 1 8 15 0 1 and 8 5 0 1 The results indicate that the double scroll attractor can be alter natively shifted to the different equilibrium point locations along the x axis with the time evolution leading to the emergence of a four double scroll chaotic attractor Setting the initial condition x 0 y 0 z 0 1 1 1 the variable x t overlapped the time domain waveforms of four level periodic pulse are simulated as shown in Fig 1a Correspondingly the phase portrait of the four double scroll attractor in the x z plane is given and plotted in Fig 1b From Fig 1 it can be seen that the proposed approach is simple and feas ible which can be used to generate multi double scroll attractor indeed J Eng 2017 doi 10 1049 joe 2017 0403 This is an open access article published by the IET under the Creative Commons Attribution License http creativecommons org licenses by 3 0 Moreover other non autonomous functions such as random bipolar sequences 7 and other different pulse like sequences can be further extended to increase the topology complexity of the gen erating chaotic attractors 3 Circuit implementation and verifi cations The circuit implementation is necessary for the verifi cations of the dynamical behaviours In order to verify the feasibility of the pro posed method a circuit schematic by using commonly off the shelf electronic components can be designed as shown in Fig 2 According to the designed circuit schematic given in Fig 2 the circuit equation can be established as RC dvx dt 10g1vyvz RC dvy dt 1 10g2v2 z RC dvz dt vx f t 10g1vyvz 5 where t tRC is the physical time vx vy and vzare the voltages of three integrating capacitors g1and g2are the gains of multipliers M1and M2 respectively f t representsthe practical four level peri odic pulse excitation It should be pointed that g1 1 10 and g2 1 10 are chosen for simplifying circuit implementation and avoiding the output saturation of the two multipliers With the circuit schematic given in Fig 2 a circuit simulation model is built by NI Multisim 12 0 simulation and circuit design software The AD711KN op amps and AD633JN multipliers with 15 V DC voltage supplies are utilised The four level periodic pulse can be implemented via series superposition of two function generators When the element parameters in Fig 2 are fi xed as R 10 k C 10 nF g1 1 10 and g2 1 10 and the amplitudes and fre quencies of two function generators are set to A1 0 533 V A2 0 266 V f1 10 Hz and f2 20 Hz respectively the time domain waveforms of the four level periodic pulse and state vari able vx t are captured by using virtual mixed signal oscilloscope Agilent 54622D as shown in Fig 3a Correspondingly the phase portrait of four double scroll chaotic attractor in the vx vz plane is displayed in Fig 3b Comparing Fig 3 with Fig 1 it has been found that the circuit simulated results are consistent well with the numerical simulations 4Conclusions In this paper a simple approach to generating multi double scroll attractor from the variable boostable chaotic system excited by non autonomous function is proposed As a paradigm a four level periodic pulse is employed as the non autonomous function to generatefour double scrollchaoticattractor Bymeansof mathematical model numerical simulations and circuit simulations are performed These results are well consistent which demonstrate the feasibility of the proposed approach Consequently the pro posed approach to generating multi double scroll attractor is simple and feasible and may be suitable for chaos based applications 5Acknowledgments This work was supported by the National Natural Science Foundation of China Grant nos 51777016 61601062 andFig 2 Designed circuit schematic for system 4 Fig 1 Numerically simulated results a Time domain waveforms of x t and f t b Phase portrait of four double scroll chaotic attractor in the x z plane Fig 3 Circuit simulated results a Time domain waveforms of vx t and f t b Phase portrait of four double scroll chaotic attractor in the vx vzplane This is an open access article published by the IET under the Creative Commons Attribution License http creativecommons org licenses by 3 0 J Eng 2017 doi 10 1049 joe 2017 0403 51607013 and the Natural Science Foundation of Jiangsu Province China Grant no BK20160282 6References 1 L J H Chen G R Generating multiscroll chaotic attractors theor ies methods and applications Int J Bifurcation Chaos 2006 16 4 pp 775 858 2 Jafari S Pham