Fractional - order chaotic systems

This contribution deals with the fractional-order chaotic systems. A survey of the chaotic systems, where total order of the system is less than 3 is presented. With using a fractional derivative a chaos can be observed in such system in spite of usual notation that chaos can occur in system with order 3 and more. A numerical method for strange attractors computation is presented as well.

[1]  Igor Podlubny,et al.  Mittag-Leffler stability of fractional order nonlinear dynamic systems , 2009, Autom..

[2]  Jun-Guo Lu,et al.  Stability Analysis of a Class of Nonlinear Fractional-Order Systems , 2008, IEEE Transactions on Circuits and Systems II: Express Briefs.

[3]  Leon O. Chua,et al.  The CNN paradigm , 1993 .

[4]  K. O’Grady,et al.  柔軟記録媒体のための金属粒子(MP)技術の開発 , 2008 .

[5]  Ivo Petrás,et al.  A Note on the Fractional-Order Cellular Neural Networks , 2006, The 2006 IEEE International Joint Conference on Neural Network Proceedings.

[6]  Philippe Deregel.  Chua's oscillator: a Zoo of attractors , 1993, J. Circuits Syst. Comput..

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

[8]  R. Rubin,et al.  Singularities of Linear System Functions , 1961 .

[9]  Xavier Moreau,et al.  An overview of the CRONE approach in system analysis, modeling and identification, observation and control , 2008 .

[10]  Hao Zhu,et al.  Chaos and synchronization of the fractional-order Chua’s system , 2009 .

[11]  Allan K. Evans,et al.  The Effects of Continuously Varying the Fractional Differential Order of Chaotic Nonlinear Systems , 1999 .

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

[13]  E. Ahmed,et al.  Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models , 2007 .

[14]  L. Greller,et al.  Explosive route to chaos through a fractal torus in a generalized lotka-volterra model , 1988 .

[15]  S. Westerlund,et al.  Capacitor theory , 1994 .

[16]  Juhn-Horng Chen,et al.  Chaotic dynamics of the fractionally damped van der Pol equation , 2008 .

[17]  D. Matignon.  Stability results for fractional differential equations with applications to control processing , 1996 .

[18]  Michael Peter Kennedy,et al.  Three steps to chaos. II. A Chua's circuit primer , 1993 .

[19]  Zheng-Ming Ge,et al.  Chaos in a fractional order modified Duffing system , 2007 .

[20]  Leon O. Chua,et al.  MEMRISTOR CELLULAR AUTOMATA AND MEMRISTOR DISCRETE-TIME CELLULAR NEURAL NETWORKS , 2009 .

[21]  Jinhu Lü,et al.  Stability analysis of linear fractional differential system with multiple time delays , 2007 .

[22]  Mohammad Saleh Tavazoei,et al.  Using fractional-order integrator to control chaos in single-input chaotic systems , 2009 .

[23]  W. Deng,et al.  Chaos synchronization of the fractional Lü system , 2005 .

[24]  Changpin Li,et al.  The synchronization of three fractional differential systems , 2007 .

[25]  Sachin Bhalekar,et al.  Chaos in fractional ordered Liu system , 2010, Comput. Math. Appl..

[26]  Jinhu Lu,et al.  A New Chaotic Attractor Coined , 2002, Int. J. Bifurc. Chaos.

[27]  Serdar Ethem Hamamci.  Stabilization using fractional-order PI and PID controllers , 2007 .

[28]  Yuan Kang,et al.  Chaos in the Newton–Leipnik system with fractional order , 2008 .

[29]  Luigi Fortuna,et al.  Fractional Order Systems: Modeling and Control Applications , 2010 .

[30]  F. Zou,et al.  Bifurcation and chaos in cellular neural networks , 1993 .

[31]  R. Leipnik,et al.  Double strange attractors in rigid body motion with linear feedback control , 1981 .

[32]  Mohammad Saleh Tavazoei,et al.  Some Applications of Fractional Calculus in Suppression of Chaotic Oscillations , 2008, IEEE Transactions on Industrial Electronics.

[33]  Ivo Petras,et al.  A note on the fractional-order Volta’s system , 2010 .

[34]  Junguo Lu,et al.  CHAOTIC DYNAMICS AND SYNCHRONIZATION OF FRACTIONAL-ORDER CHUA'S CIRCUITS WITH A PIECEWISE-LINEAR NONLINEARITY , 2005 .

[35]  M. Najjari,et al.  New modeling of the power diode using the VHDL-AMS language , 2008 .

[36]  S. Westerlund.  Dead matter has memory , 1991 .

[37]  Alberto Tesi,et al.  Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems , 1992, Autom..

[38]  Wei-Ching Chen,et al.  Nonlinear dynamics and chaos in a fractional-order financial system , 2008 .

[39]  M. Nakagawa,et al.  Basic Characteristics of a Fractance Device , 1992 .

[40]  L. Chua,et al.  The double scroll family , 1986 .

[41]  D. Matignon,et al.  Some Results on Controllability and Observability of Finite-dimensional Fractional Differential Systems , 1996 .

[42]  Takashi Matsumoto,et al.  A chaotic attractor from Chua's circuit , 1984 .

[43]  Michael Peter Kennedy,et al.  Three steps to chaos. I. Evolution , 1993 .

[44]  C. Halijak,et al.  Approximation of Fractional Capacitors (1/s)^(1/n) by a Regular Newton Process , 1964 .

[45]  Naresh K. Sinha,et al.  Modern Control Systems , 1981, IEEE Transactions on Systems, Man, and Cybernetics.

[46]  J. C. Wang.  Realizations of Generalized Warburg Impedance with RC Ladder Networks and Transmission Lines , 1987 .

[47]  Leon O. Chua,et al.  An IC chip of Chua's circuit , 1993 .

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

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

[50]  Mohammad Saleh Tavazoei,et al.  Chaotic attractors in incommensurate fractional order systems , 2008 .

[51]  I. Schäfer,et al.  Modelling of lossy coils using fractional derivatives , 2008 .

[52]  M. Haeri,et al.  Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems , 2007 .

[53]  Tao Liu,et al.  A novel three-dimensional autonomous chaos system , 2009 .

[54]  Neville J. Ford,et al.  The numerical solution of fractional differential equations: Speed versus accuracy , 2001, Numerical Algorithms.

[55]  Chunguang Li,et al.  Chaos and hyperchaos in the fractional-order Rössler equations , 2004 .

[56]  W. M. Stone.  Complex Variables and the Laplace Transform for Engineers (W. R. LePage) , 1961 .

[57]  L. Chua.  Memristor-The missing circuit element , 1971 .

[58]  Yangquan Chen,et al.  Two direct Tustin discretization methods for fractional-order differentiator/integrator , 2003, J. Frankl. Inst..

[59]  Lu Jun-Guo,et al.  Chaotic dynamics and synchronization of fractional-order Genesio–Tesi systems , 2005 .

[60]  Guanrong Chen,et al.  Chen's Attractor Exists , 2004, Int. J. Bifurc. Chaos.

[61]  Chai Wah Wu,et al.  Chua's oscillator: A compendium of chaotic phenomena , 1994 .

[62]  Le Page,et al.  Complex Variables and the Laplace Transform for Engineers , 2010 .

[63]  L. Chua,et al.  The double hook (nonlinear chaotic circuits) , 1988 .

[64]  Alain Oustaloup,et al.  Frequency-band complex noninteger differentiator: characterization and synthesis , 2000 .

[65]  A. Oustaloup.  La dérivation non entière , 1995 .

[66]  Michael Peter Kennedy,et al.  Robust OP Amp Realization of Chua's Circuit , 1992 .

[67]  Z. Ge,et al.  Chaos in a generalized van der Pol system and in its fractional order system , 2007 .

[68]  L. Dorcak.  Numerical Models for the Simulation of the Fractional-Order Control Systems , 2002 .

[69]  Juebang Yu,et al.  Chaos in the fractional order periodically forced complex Duffing’s oscillators , 2005 .

[70]  P. Arena,et al.  Bifurcation and Chaos in Noninteger Order Cellular Neural Networks , 1998 .

[71]  I. Petráš.  Control of Fractional-Order Chua's System , 2000, nlin/0008029.

[72]  L. Chua,et al.  A universal circuit for studying and generating chaos. I. Routes to chaos , 1993 .

[73]  J. A. Tenreiro Machado,et al.  Analysis of the Van der Pol Oscillator Containing Derivatives of Fractional Order , 2007 .

[74]  Michael Peter Kennedy.  Three Steps to Chaos-Part I: Evolution , 1993 .

[75]  Ivo Petras,et al.  A note on the fractional-order Chua’s system , 2008 .

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

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

[78]  George M. Zaslavsky.  Hamiltonian Chaos and Fractional Dynamics , 2005 .

[79]  Yury F. Luchko,et al.  Algorithms for the fractional calculus: A selection of numerical methods , 2005 .

[80]  G. Zhong.  Implementation of Chua's circuit with a cubic nonlinearity , 1994 .

[81]  Weihua Deng,et al.  Short memory principle and a predictor-corrector approach for fractional differential equations , 2007 .

[82]  Jun-Guo Lu,et al.  Chaotic dynamics and synchronization of fractional-order Arneodo’s systems , 2005 .

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

[84]  B. Hao,et al.  Elementary Symbolic Dynamics And Chaos In Dissipative Systems , 1989 .

[85]  I. Podlubny.  Fractional differential equations , 1998 .

[86]  P. Arena,et al.  Nonlinear Noninteger Order Circuits and Systems — An Introduction , 2000 .

[87]  Mohammad Saleh Tavazoei,et al.  A note on the stability of fractional order systems , 2009, Math. Comput. Simul..

[88]  Marco Gilli,et al.  Bifurcations And Chaos In Cellular Neural Networks , 2003, J. Circuits Syst. Comput..

[89]  Mohammad Saleh Tavazoei,et al.  Chaos control via a simple fractional-order controller , 2008 .

[90]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[91]  D. Matignon.  Stability properties for generalized fractional differential systems , 1998 .

[92]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .