Regular oscillations or chaos in a fractional order system with any effective dimension

Abstract This paper introduces a fractional order system which can generate regular oscillations or create chaos. It shows that this system is capable to create regular or nonregular oscillations under suitable conditions. These necessary conditions are achieved by violation of the no-chaos criteria. The effective dimension of the proposed system can be chosen any order less than three. Therefore, this system is a good example for limit cycle or chaos generation via fractional-order systems with low orders. Numerical simulations illustrate behavior of the proposed system in different situations.

[1]  Junguo Lu.  Chaotic dynamics of the fractional-order Lü system and its synchronization , 2006 .

[2]  Wajdi M. Ahmad.  Generation and control of multi-scroll chaotic attractors in fractional order systems , 2005 .

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

[4]  S. Natsiavas,et al.  Periodic response and stability of oscillators with symmetric trilinear restoring force , 1989 .

[5]  K. Cole.  ELECTRIC CONDUCTANCE OF BIOLOGICAL SYSTEMS , 1933 .

[6]  Guanrong Chen,et al.  Generating chaos with a switching piecewise-linear controller. , 2002, Chaos.

[7]  Changpin Li,et al.  Does the fractional Brusselator with efficient dimension less than 1 have a limit cycle , 2007 .

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

[9]  K. B. Oldham,et al.  Analogue instrumentation for processing polarographic data , 1983 .

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

[11]  Marina V. Shitikova,et al.  Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems , 1997 .

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

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

[14]  S. O. Scanlan,et al.  Synthesis of piecewise-linear chaotic oscillators with prescribed eigenvalues , 2001 .

[15]  James Gleick.  Chaos: Making a New Science , 1987 .

[16]  A. Méhauté,et al.  Introduction to transfer and motion in fractal media: The geometry of kinetics , 1983 .

[17]  S. J. Linz.  No-chaos criteria for certain jerky dynamics , 2000 .

[18]  J. Yorke,et al.  Period Three Implies Chaos , 1975 .

[19]  卢俊国,et al.  Chaotic dynamics of the fractional-order Ikeda delay system and its synchronization , 2006 .

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

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

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

[23]  A. El-Sayed,et al.  Fractional-order diffusion-wave equation , 1996 .

[24]  F. Busse.  An exploration of chaos: J. Argyris, G. Faust and M. Haase, Elsevier, Amsterdam, 1994, 722 pp., ISBN 0-444-82002-7 (hardbound), 0-444-82003-5 (paperback) , 1994 .

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

[26]  P. Holmes,et al.  A periodically forced piecewise linear oscillator , 1983 .

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

[28]  J.A.T. Machado,et al.  Dynamics of the fractional-order Van der Pol oscillator , 2004, Second IEEE International Conference on Computational Cybernetics, 2004. ICCC 2004..

[29]  Johan A. K. Suykens,et al.  Families of scroll Grid attractors , 2002, Int. J. Bifurc. Chaos.

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

[31]  Victor George Jenson,et al.  Mathematical Methods in Chemical Engineering , 1978 .

[32]  James Gleick,et al.  Chaos, Making a New Science , 1987 .

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

[34]  Henry Leung,et al.  Design of 3-D multi-scroll chaotic attractors via basic circuits , 2005 .

[35]  Guoqing Chen,et al.  An RLC interconnect model based on fourier analysis , 2005, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[36]  N. Engheta.  On fractional calculus and fractional multipoles in electromagnetism , 1996 .

[37]  Guanrong Chen,et al.  Generating two simultaneously chaotic attractors with a switching piecewise-linear controller , 2004 .

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

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

[40]  Jinhu Lu,et al.  Design and implementation of multi-directional grid multi-torus chaotic attractors , 2006, 2006 IEEE International Symposium on Circuits and Systems.

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

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

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

[44]  Xiao-Song Yang,et al.  Chaotic Attractor in a Simple Hybrid System , 2002, Int. J. Bifurc. Chaos.

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

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

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

[48]  Jinchen Ji,et al.  Dynamics of a piecewise linear system subjected to a saturation constraint , 2004 .

[49]  Elena Grigorenko,et al.  Erratum: Chaotic Dynamics of the Fractional Lorenz System [Phys. Rev. Lett.91, 034101 (2003)] , 2006 .

[50]  Weihua Deng,et al.  Design of multidirectional multiscroll chaotic attractors based on fractional differential systems via switching control. , 2006, Chaos.

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

[52]  Julien Clinton Sprott,et al.  Elementary chaotic flow , 1999 .

[53]  Thomas J. Anastasio,et al.  The fractional-order dynamics of brainstem vestibulo-oculomotor neurons , 1994, Biological Cybernetics.