Analysis and circuit implementation for the fractional-order Lorenz system

Transfer function approximation in frequency domain is not only one of common numerical analysis methods studying portraits of fractional-order chaotic systems, but also a main method to design their chaotic circuits. According to it, in this paper we first investigate the chaotic characteristics of the fractional-order Lorenz system, find some more complex dynamics by analyzing Lyapunov exponents diagrams, bifurcation diagrams and phase portraits, that is, we display the chaotic characteristics as well as periodic characteristics of the system when changing fractional-order from 0.7 to 0.9 in steps of 0.1, and show that the chaotic motion exists in the a lower-dimensional fractional-order Lorenz system. Then, according to transfer function approximation and the approach to designing integer-order chaotic circuits, we also design an analog circuit to implement the fractional-order system. The resistors and capacitors in the circuit are selected according to the system parameters and transfer function approximation in frequency domain. Some phase portraits including chaotic attractors and periodic attractors are observed by oscilloscope, which are coincident well with numerical simulations, and the chaotic characteristics of the fractional-order Lorenz system are further proved by the physical implementation.

[1]  E. Selçuk,et al.  Control of a chaotic finance system with passive control , 2012 .

[2]  Guoliang Cai,et al.  Stabilization of a Modified Chaotic Finance System , 2011, 2011 Fourth International Conference on Information and Computing.

[3]  P. Linsay.  Period Doubling and Chaotic Behavior in a Driven Anharmonic Oscillator , 1981 .

[4]  Guanrong Chen,et al.  A note on the fractional-order Chen system , 2006 .

[5]  Adi Soeprijanto,et al.  Controlling chaos and voltage collapse using an ANFIS-based composite controller-static var compensator in power systems , 2013 .

[6]  Recai Kiliç.  Experimental Study of Cfoa-Based inductorless Chua's Circuit , 2004, Int. J. Bifurc. Chaos.

[7]  Hiroyuki Kitajima,et al.  Bifurcations in a Coupled Rossler System , 1995 .

[8]  J. C. Sprotta.  Simple chaotic systems and circuits , 2000 .

[9]  Guoliang Cai,et al.  Dynamic analysis and control of a new hyperchaotic finance system , 2012 .

[10]  Michael Peter Kennedy.  Chaos in the Colpitts oscillator , 1994 .

[11]  Hongyan Jia,et al.  Topological horseshoe analysis and the circuit implementation for a four-wing chaotic attractor , 2011 .

[12]  Tiedong Ma,et al.  Cluster synchronization in fractional-order complex dynamical networks , 2012 .

[13]  T. Carroll.  A simple circuit for demonstrating regular and synchronized chaos , 1995 .

[14]  Ahmed S. Elwakil,et al.  A family of Wien-type oscillators modified for chaos , 1997, Int. J. Circuit Theory Appl..

[15]  Jianfeng Wang,et al.  Globally Exponentially Attractive Set and Synchronization of a Class of Chaotic Finance System , 2009, ISNN.

[16]  M. H. Purnomo,et al.  Controlling chaos using ANFIS-based Composite Controller (ANFIS-CC) in power systems , 2009, International Conference on Instrumentation, Communication, Information Technology, and Biomedical Engineering 2009.

[17]  S. Čelikovský,et al.  Control systems: from linear analysis to synthesis of chaos , 1996 .

[18]  Y. Uyaroglu,et al.  A new chaotic attractor from general Lorenz system family and its electronic experimental implementation , 2010 .

[19]  Lixia Liu,et al.  Hopf bifurcation and chaos of financial system on condition of specific combination of parameters* , 2008, J. Syst. Sci. Complex..

[20]  Recai Kiliç,et al.  Experimental Verification of High Frequency Performance of FTFN-Based Simple Chaotic Circuit , 2005, Int. J. Bifurc. Chaos.

[21]  B. Onaral,et al.  Fractal system as represented by singularity function , 1992 .

[22]  Mohammad Saleh Tavazoei,et al.  Limitations of frequency domain approximation for detecting chaos in fractional order systems , 2008 .

[23]  Juan Ding,et al.  A New Modified Hyperchaotic Finance System and its Control , 2009 .

[24]  Ma Jun-hai,et al.  Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system(II) , 2001 .

[25]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[26]  O. Rössler.  An equation for continuous chaos , 1976 .

[27]  Zhenbo Li,et al.  Synchronization of a chaotic finance system , 2011, Appl. Math. Comput..

[28]  C. Wu,et al.  A Universal Circuit for Studying and Generating Chaos-Part I: Routes , 1993 .

[29]  S. Tong,et al.  Adaptive fuzzy approach to control unified chaotic systems , 2007 .

[30]  Ahmad Bagheri,et al.  Decoupled adaptive neuro-fuzzy (DANF) sliding mode control system for a Lorenz chaotic problem , 2009, Expert Syst. Appl..

[31]  Shih-Yu Li.  Chaos control of new Mathieu-van der Pol systems by fuzzy logic constant controllers , 2011, Appl. Soft Comput..

[32]  Xingyuan Wang,et al.  Hopf bifurcation and topological horseshoe of a novel finance chaotic system , 2012 .

[33]  Guoliang Cai,et al.  Chaos Control of a Non-linear Finance System , 2011 .

[34]  Ivo Petráš,et al.  Chaos in the fractional-order Volta’s system: modeling and simulation , 2009 .

[35]  Guoliang Cai,et al.  A New Finance Chaotic Attractor , 2007 .

[36]  K S OJO,et al.  Function projective synchronization of identical and non-identical modified finance and Shimizu–Morioka systems , 2012 .

[37]  Oliver Heaviside,et al.  Electromagnetic theory : Including an account of Heaviside's unpublished notes for a fourth volume and with a foreword by Edmund Whittaker , 1971 .

[38]  Julien Clinton Sprott,et al.  Chaos in fractional-order autonomous nonlinear systems , 2003 .

[39]  Junhai Ma,et al.  Chaos and Hopf bifurcation of a finance system , 2009 .

[40]  Wajdi M. Ahmad.  Hyperchaos in fractional order nonlinear systems , 2005 .

[41]  Uyaroğlu Yılmaz.  Feedback control of chaos in a hyperchaotic finance system , 2012 .

[42]  Recai Kiliç,et al.  Chaos training boards: Versatile pedagogical tools for teaching chaotic circuits and systems , 2008 .

[43]  Mohammad Saleh Tavazoei,et al.  A necessary condition for double scroll attractor existence in fractional-order systems , 2007 .

[44]  Wei-Ching Chen,et al.  Dynamics and control of a financial system with time-delayed feedbacks , 2008 .

[45]  C. F. Lorenzo,et al.  Chaos in a fractional order Chua's system , 1995 .

[46]  Ma Junhai,et al.  Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (I) , 2001 .

[47]  Elena Grigorenko,et al.  Chaotic dynamics of the fractional Lorenz system. , 2003, Physical review letters.

[48]  Guoliang Cai,et al.  Globally Exponentially Attractive Set and Synchronization of a Novel Three-Dimensional Chaotic Finance System , 2010, 2010 Third International Conference on Information and Computing.

[49]  A. Tamasevicius,et al.  Modified Wien-bridge oscillator for chaos , 1995 .