This paper deals with the problem of determining the conditions under which fractional order R？ssler toroidal system can give rise to chaotic behavior. Based on the harmonic balance method, four detailed steps are presented for predicting the existence and the location of chaotic motions. Numerical simulations are performed to verify the theoretical analysis by straightforward computations. 1. Introduction The concept of fractional calculus has been known since the development of the regular calculus, with the first reference probably being associated with Leibniz and L’H？spital in 1695. In the past three decades or so, fractional calculus gained considerable popularity and importance, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering [1–3]. In particular, by utilizing fractional calculus technique, many investigations were devoted to the chaotic behaviors and chaotic control of dynamical systems involving the fractional derivative, called fractional-order chaotic system [4–8]. For example, it has been shown that Chua circuit of the order as low as 2.7 can behave in chaotic manner [4]. In [5], the nonautonomous Duffing systems of the order less than 2 can still produce a chaotic attractor. In [6], chaotic behavior of the fractional-order Lorenz system was further studied. In [7], chaos and hyperchaos in the fractional-order R？ssler equations were also studied, in which the authors showed that chaos can exist in the fractional-order R？ssler equation with the order as low as 2.4, and hyperchaos can exist in the fractional-order R？ssler hyperchaos equation with the order as low as 3.8. Later on, the chaotic behavior and its control in the fractional-order Chen system were investigated in [8]. And recently, more dynamic behaviors of fractional order chaotic systems were analyzed by using different approaches; we refer the readers to [9–12]. However, to our knowledge, the conditions for chaos existence in dynamical systems (including integer-order system and fractional-order system) are still incomplete. For a given dynamical system, can we decide (without invoking numerical simulations) whether and in what parameter ranges, chaotic behavior might exist? It is still an open problem [13]. Even though some theorems such as Melnikov’s criteria [14] and Shil’nikov’s theorem [15] may be helpful in some special cases, it seems that a powerful tool is not generally available to determine the accurate parameter ranges for chaos existence in a given dynamical system. Moreover, the fact that
References
[1]
K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, London, UK, 1974.
[2]
I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press Inc., San Diego, Calif, USA, 1999.
[3]
R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000.
[4]
T. T. Hartley, C. F. Lorenzo, and H. K. Qammer, “Chaos in a fractional order Chua's system,” IEEE Transactions on Circuits and Systems I, vol. 42, no. 8, pp. 485–490, 1995.
[5]
P. Arena, R. Caponetto, L. Fortuna, and D. Porto, “Chaos in a fractional order Duffing system,” in Proceedings of European Conference on Circuit Theory and Design, pp. 1259–1262, Budapest, Hungary, 1997.
[6]
I. Grigorenko and E. Grigorenko, “Chaotic dynamics of the fractional Lorenz system,” Physical Review Letters, vol. 91, no. 3, Article ID 034101, 2003.
[7]
C. G. Li and G. R. Chen, “Chaos and hyperchaos in the fractional-order R？ssler equations,” Physica A, vol. 341, no. 1–4, pp. 55–61, 2004.
[8]
C. P. Li and G. R. Chen, “Chaos in the fractional order Chen system and its control,” Chaos, Solitons and Fractals, vol. 22, no. 3, pp. 549–554, 2004.
[9]
V. Daftardar-Gejji and S. Bhalekar, “Chaos in fractional ordered Liu system,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1117–1127, 2010.
[10]
Q. Yang and C. Zeng, “Chaos in fractional conjugate Lorenz system and its scaling attractors,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 4041–4051, 2010.
[11]
C. Zeng, Q. Yang, and J. Wang, “Chaos and mixed synchronization of a new fractional-order system with one saddle and two stable node-foci,” Nonlinear Dynamics, vol. 65, no. 4, pp. 457–466, 2011.
[12]
Z. Wei, “Synchronization of coupled nonidentical fractional-order hyperchaotic systems,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 430724, 9 pages, 2011.
[13]
S. J. Linz, “No-chaos criteria for certain classes of driven nonlinear oscillators,” Acta Physica Polonica B, vol. 34, no. 7, pp. 3741–3749, 2003.
[14]
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42, Springer, New York, NY, USA, 1983.
[15]
C. P. Silva, “Shil’nikov’s theorem—a tutorial,” IEEE Transactions on Circuits and Systems I, vol. 40, no. 10, pp. 675–682, 1993.
[16]
Z. M. Wu, J. G. Lu, and J. Y. Xie, “Analysing chaos in fractional-order systems with the harmonic balance method,” Chinese Physics, vol. 15, no. 6, pp. 1201–1207, 2006.
[17]
M. S. Tavazoei and M. Haeri, “Describing function based methods for predicting chaos in a class of fractional order differential equations,” Nonlinear Dynamics, vol. 57, no. 3, pp. 363–373, 2009.
[18]
K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002.
[19]
K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 3–22, 2002.
[20]
O. E. R？ssler, “Continuous chaos—four prototype equations,” Annals of the New York Academy of Sciences, vol. 316, pp. 376–392, 1979.
[21]
K. Ciesielski, “On the Poincare-Bendixson theorem,” in Lecture Notes in Nonlinear Analysis, vol. 3, pp. 49–69, 2002.
[22]
R. Genesio and A. Tesi, “Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems,” Automatica, vol. 28, no. 3, pp. 531–548, 1992.
[23]
A. Gelb and W. E. Velde, Multiple-Input Describing Functions and Nonlinear System Design, McGraw Hill, New York, NY, USA, 1967.
[24]
F. A. Savaci and S. Günel, “Harmonic balance analysis of the generalized Chua's circuit,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, no. 8, pp. 2325–2332, 2006.
[25]
C. Bonnet and J. R. Partington, “Coprime factorizations and stability of fractional differential systems,” Systems & Control Letters, vol. 41, no. 3, pp. 167–174, 2000.
[26]
R. Genesio, A. Tesi, and F. Villoresi, “Frequency approach for analyzing and controlling chaos in nonlinear circuits,” IEEE Transactions on Circuits and Systems I, vol. 40, no. 11, pp. 819–828, 1993.
