What is the lowest order of the fractional-order chaotic systems to behave chaotically?

Abstract Considering the fractional-order simplified Lorenz system as an example, we analyze the lowest order under the effects of variations in the numerical methods employed to solve the system. We varied the commensurability of the system equations, the system parameter and iteration step to reveal how they affect the appearance of chaos. The results show that the lowest order is obtained using the Adomian decomposition method (ADM), and it is smaller than that obtained through the Adams-Bashforth-Moulton (ABM) algorithm. The fractional-order system in the case of incommensurate order always has a smaller lowest order for chaos than does the system with commensurate order. We found that the lowest order of the fractional-order system decreases as the system parameter increases or as the iteration step decreases. In addition, by selecting the optimal results from a previous analysis, we obtain the lowest order of the fractional-order simplified Lorenz system is 1.136, which is lower than all results previously reported. Finally, we propose a corollary: the lowest order of a fractional-order chaotic system returned by an exact solution condition is lower than the approximate solution derived by numerical simulations. This work facilitates us to find a lower lowest order of fractional-order nonlinear systems, which are relevant to the study of fractional-order nonlinear system for engineering application.

[1]  Giuseppe Grassi,et al.  Chaos in the fractional Chua and Chen systems with lowest-order , 2008, 2008 15th IEEE International Conference on Electronics, Circuits and Systems.

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

[3]  R. Gorenflo,et al.  Fractional Calculus: Integral and Differential Equations of Fractional Order , 2008, 0805.3823.

[4]  Georg A. Gottwald,et al.  On the Implementation of the 0-1 Test for Chaos , 2009, SIAM J. Appl. Dyn. Syst..

[5]  Jafar Biazar,et al.  A Simple Algorithm for Calculating Adomian Polynomials , 2007 .

[6]  Julien Clinton Sprott,et al.  Bifurcations and Chaos in fractional-Order Simplified Lorenz System , 2010, Int. J. Bifurc. Chaos.

[7]  G. Adomian.  A new approach to nonlinear partial differential equations , 1984 .

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

[9]  Riccardo Caponetto,et al.  An Application of Adomian Decomposition for Analysis of fractional-Order Chaotic Systems , 2013, Int. J. Bifurc. Chaos.

[10]  Kehui Sun,et al.  Numerical analysis of a fractional-order chaotic system based on conformable fractional-order derivative , 2017 .

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

[12]  Hossein Jafari,et al.  Adomian decomposition: a tool for solving a system of fractional differential equations , 2005 .

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

[14]  Chunguang Li,et al.  Chaos in the fractional order Chen system and its control , 2004 .

[15]  Kamel Al-khaled,et al.  Numerical solutions for systems of fractional differential equations by the decomposition method , 2005, Appl. Math. Comput..

[16]  Varsha Daftardar-Gejji,et al.  On disappearance of chaos in fractional systems , 2017 .

[17]  Giuseppe Grassi,et al.  Bifurcation and Chaos in the fractional-Order Chen System via a Time-Domain Approach , 2008, Int. J. Bifurc. Chaos.

[18]  Kehui Sun,et al.  Dynamics of a Simplified Lorenz System , 2009, Int. J. Bifurc. Chaos.

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

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

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

[22]  José António Tenreiro Machado,et al.  What is a fractional derivative? , 2015, J. Comput. Phys..

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

[24]  S. Samko,et al.  Fractional integrals and derivatives: mapping properties , 2016 .

[25]  Kehui Sun,et al.  Solution and dynamics of a fractional-order 5-D hyperchaotic system with four wings , 2017 .

[26]  Jian-Feng Zhao,et al.  A novel image encryption scheme based on an improper fractional-order chaotic system , 2015, Nonlinear Dynamics.

[27]  M. Rivero,et al.  Fractional calculus: A survey of useful formulas , 2013, The European Physical Journal Special Topics.

[28]  Nazlı Yazıcı Gözütok,et al.  Multi-variable conformable fractional calculus , 2018 .

[29]  王会海,et al.  Solution of the fractional-order chaotic system based on Adomian decomposition algorithm and its complexity analysis , 2014 .

[30]  Jie Li,et al.  Chaos in the fractional order unified system and its synchronization , 2008, J. Frankl. Inst..

[31]  Kehui Sun,et al.  A novel color image encryption scheme using fractional-order hyperchaotic system and DNA sequence operations* , 2017 .

[32]  Firdaus E. Udwadia,et al.  An efficient QR based method for the computation of Lyapunov exponents , 1997 .

[33]  Cheng Jiaxing,et al.  Color Image Encryption Algorithm Based on DNA Sequence Operation and Fractional Order Chen Hyper-Chaotic System , 2016 .