Chaos in the Newton–Leipnik system with fractional order

The dynamics of fractional-order systems has attracted increasing attention in recent years. In this paper, the dynamics of the Newton–Leipnik system with fractional order was studied numerically. The system displays many interesting dynamic behaviors, such as fixed points, periodic motions, chaotic motions, and transient chaos. It was found that chaos exists in the fractional-order system with order less than 3. In this study, the lowest order for this system to yield chaos is 2.82. A period-doubling route to chaos in the fractional-order system was also found.

[1]  Ahmed S. Elwakil,et al.  Fractional-order Wien-bridge oscillator , 2001 .

[2]  Hsien-Keng Chen,et al.  Anti-control of chaos in rigid body motion , 2004 .

[3]  Chyi Hwang,et al.  A note on time-domain simulation of feedback fractional-order systems , 2002, IEEE Trans. Autom. Control..

[4]  Zheng-Ming Ge,et al.  Stability and Chaotic Motions of a Symmetric Heavy Gyroscope. , 1996 .

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

[6]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .

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

[8]  Hendrik Richter Controlling chaotic systems with multiple strange attractors , 2002 .

[9]  Z. Ge,et al.  THE REGULAR AND CHAOTIC MOTIONS OF A SYMMETRIC HEAVY GYROSCOPE WITH HARMONIC EXCITATION , 1996 .

[10]  H.-K. Chen CHAOS AND CHAOS SYNCHRONIZATION OF A SYMMETRIC GYRO WITH LINEAR-PLUS-CUBIC DAMPING , 2002 .

[11]  Hsien-Keng Chen,et al.  Chaotic dynamics of the fractionally damped Duffing equation , 2007 .

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

[13]  H.-K. Chen,et al.  Synchronization of chaotic symmetric gyros by one-way coupling conditions , 2003 .

[14]  G. Zaslavsky Chaos, fractional kinetics, and anomalous transport , 2002 .

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

[16]  N. Laskin Fractional market dynamics , 2000 .

[17]  R. Hilfer Applications Of Fractional Calculus In Physics , 2000 .

[18]  R. Koeller Applications of Fractional Calculus to the Theory of Viscoelasticity , 1984 .

[19]  D. Kusnezov,et al.  Quantum Levy Processes and Fractional Kinetics , 1999, chao-dyn/9901002.

[20]  T. Hartley,et al.  Dynamics and Control of Initialized Fractional-Order Systems , 2002 .

[21]  X. Tong,et al.  Chaotic Motion of a Symmetric Gyro Subjected to a Harmonic Base Excitation , 2001 .

[22]  M. Caputo Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[23]  K. Diethelm AN ALGORITHM FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER , 1997 .

[24]  Lixin Tian,et al.  Bifurcation analysis and linear control of the Newton–Leipnik system , 2006 .

[25]  B. Onaral,et al.  Linear approximation of transfer function with a pole of fractional power , 1984 .

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

[27]  M. Ichise,et al.  An analog simulation of non-integer order transfer functions for analysis of electrode processes , 1971 .

[28]  P. Arena,et al.  Chaotic behavior in noninteger-order cellular neural networks , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[29]  R. Koeller Polynomial operators, stieltjes convolution, and fractional calculus in hereditary mechanics , 1986 .

[30]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[31]  Changpin Li,et al.  Chaos in Chen's system with a fractional order , 2004 .

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

[33]  R. Bagley,et al.  Fractional order state equations for the control of viscoelasticallydamped structures , 1991 .

[34]  I. Podlubny,et al.  Analogue Realizations of Fractional-Order Controllers , 2002 .