-
摘要: 利用最近提出的广义分数阶微积分算子，研究了一类新的带Caputo型导数的广义分数阶混沌系统.讨论了混沌性质对导数阶数与系统参数的依赖性，利用有限差分法对广义分数阶混沌系统进行数值模拟，结果显示广义分数阶系统不仅蕴含经典混沌系统的结果，还展现出其他的动力学行为.由于广义分数阶混沌系统统一了多个不同的系统，未来该领域有望获得更进一步的研究.Abstract: In view of a new generalized fractional calculus proposed recently, this paper is devoted to applying the generalized fractional derivatives to study new generalized fractional chaotic systems. The chaotic properties depending on the new generalized fractional derivative are discussed and shown graphically. The generalized fractional derivative is described in the Caputo sense, and the finite difference approach for solving the generalized fractional chaotic system is presented. Since the generalized fractional derivative includes many existing fractional derivatives as special cases, we hope more attention will be brought into this field in the near future.
-
Key words:
- Finite difference method /
- fractional calculus /
- fractional chaotic system /
- generalized fractional derivative /
- nonlinear dynamics
-
Fig. 1 Phase portraits of GFLVS (top row, (a) and (b)) and GFLS (bottom row, (c) and (d)).
Fig. 2 Influence of scale function $\sigma(t)$ on GFLVS (top row, (a) and (b)) and GFLS (bottom row, (c) and (d)).
-
[1] N. Laskin, "Fractional quantum mechanics, "Phys. Rev. E, vol. 62, no. 3, pp. 3135-3145, Sep. 2000. http://www.ncbi.nlm.nih.gov/pubmed/11088808 [2] R. C. Koeller, "Applications of fractional calculus to the theory of viscoelasticity, "J. Appl. Mech. , vol. 51, no. 2, pp. 299-307, Jun. 1984. https://academic.oup.com/ptp [3] I. S. Jesus, J. A. T. Machado, and J. B. Cunha, "Fractional electrical impedances in botanical elements, "J. Vib. Control, vol. 14, no. 9-10, pp. 1389-1402, Sep. 2008. https://www.researchgate.net/publication/237464575_Fractional_Electrical_Impedances_in_Botanical_Elements [4] K. Oldham, "Fractional differential equations in electrochemistry, "Adv. Eng. Softw. , vol. 41, no. 1, pp. 9-12, Jan. 2010. http://dl.acm.org/citation.cfm?id=1645639 [5] R. L. Magin, "Fractional calculus models of complex dynamics in biological tissues, "Comput. Math. Appl. , vol. 59, no. 5, pp. 1586-1593, Mar. 2010. http://www.sciencedirect.com/science/article/pii/S0898122109005343 [6] R. L. Magin, Fractional Calculus in Bioengineering. Ridgefield, Conn, USA: Begell House Press, 2006. [7] B. Mathieu, P. Melchior, A. Oustaloup, and C. Ceyral, "Fractional differentiation for edge detection, "Signal Process. , vol. 83, no. 11, pp. 2421-2432, Nov. 2003. http://dl.acm.org/citation.cfm?id=958047 [8] D. Y. Xue and Y. Q. Chen. "Fractional order calculus and its applications in mechatronic system controls organizers, "in Proc. 2006 IEEE Int. Conf. Mechatronics and Automation, Luoyang, Henan, China, 2006, pp. nil33-nil33. http://ieeexplore.ieee.org/xpls/icp.jsp?arnumber=4026037 [9] C. Li and G. Chen, "Chaos in the fractional order Chen system and its control, "Chaos, Solitons Fractals, vol. 22, no. 3, pp. 549-554, Nov. 2004. http://www.sciencedirect.com/science/article/pii/S0960077904001250 [10] M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, "Existence results for fractional functional differential inclusions with infinite delay and applications to control theory, "Fract. Calc. Appl. Anal. , vol. 11, no. 1, pp. 35-56, 2008. http://www.sciencedirect.com/science/article/pii/S0003448708001546 [11] B. M. Vinagre, I. Podlubny, A. Hernndez, and V. Feliu, "Some approximations of fractional order operators used in control theory and applications, "Fract. Calc. Appl. Anal. , vol. 3, no. 3, pp. 231-248, Jan. 2000. http://www.ams.org/mathscinet-getitem?mr=1788163 [12] K. Diethelm, The Analysis of Fractional Differential Equations. Berlin Heidelberg, Germany: Springer, 2010. http://adsabs.harvard.edu/abs/2013OPhy...11..824X [13] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier, 2006. http://www.ams.org/mathscinet-getitem?mr=2218073 [14] I. Podlubny, Fractional Differential Equations. San Diego, USA: Academic Press, 1999. [15] Y. F. Xu and Z. M. He, "Synchronization of variable-order fractional financial system via active control method, "Cent. Eur. J. Phys. , vol. 11, no. 6, pp. 824-835, Jun. 2013. http://adsabs.harvard.edu/abs/2013OPhy...11..824X [16] M. Asheghan, S. Delshad, B. Hamidi, and M. Tavazoei, "Non-fragile control and synchronization of a new fractional order chaotic system, "Appl. Math. Comput. , vol. 222, pp. 712-721, Oct. 2013. http://dl.acm.org/citation.cfm?id=2621316 [17] D. Y. Chen, R. F. Zhang, J. C. Sprott, and X. Y. Ma, "Synchronization between integer-order chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode control, "Nonlinear Dyn. , vol. 70, no. 2, pp. 1549-1561, Oct. 2012. doi: 10.1007/s11071-012-0555-3 [18] H. Q. Li, X. F. Liao, and M. W. Luo, "A novel non-equilibrium fractional-order chaotic system and its complete synchronization by circuit implementation, "Nonlinear Dyn. , vol. 68, no. 1, pp. 137-149, Apr. 2012. doi: 10.1007/s11071-011-0210-4 [19] D. Cafagna and G. Grassi, "Observer-based projective synchronization of fractional systems via a scalar signal: Application to hyperchaotic Rössler systems, "Nonlinear Dyn. , vol. 68, no. 1, pp. 117-128, Apr. 2012. http://www.ams.org/mathscinet-getitem?mr=2904161 [20] V. S. Kiryakova, Generalized Fractional Calculus and Applications. Harlow, Essex, England, New York, USA: CRC Press, 1993. [21] O. P. Agrawal, "Generalized variational problems and Euler-Lagrange equations, "Comput. Math. Appl. , vol. 59, no. 5, pp. 1852-1864, Mar. 2010. http://dl.acm.org/citation.cfm?id=1746947 [22] O. P. Agrawal, "Some generalized fractional calculus operators and their applications in integral equations, "Fract. Calc. Appl. Anal. , vol. 15, no. 4, pp. 700-711, Dec. 2012. doi: 10.2478/s13540-012-0047-7 [23] Y. F. Xu, Z. M. He, and Q. W. Xu, "Numerical solutions of fractional advection-diffusion equations with a kind of new generalized fractional derivative, "Int. J. Comput. Math. , vol. 91, no. 3, pp. 588-600, Mar. 2014. doi: 10.1080/00207160.2013.799277 [24] Y. F. Xu and O. P. Agrawal, "Numerical solutions and analysis of diffusion for new generalized fractional Burgers equation, "Fract. Calc. Appl. Anal. , vol. 16, no. 3, pp. 709-736, Sep. 2013. doi: 10.2478/s13540-013-0045-4 [25] Xu Y, He Z, and Agrawal O, "Numerical and analytical solutions of new generalized fractional diffusion equation, "Comput. Math. Appl. , vol. 66, no. 10, pp. 2019-2029, Dec. 2013. http://dl.acm.org/citation.cfm?id=2542944 [26] Y. F. Xu and O. P. Agrawal, "Numerical solutions and analysis of diffusion for new generalized fractional advection-diffusion equations, "Cent. Eur. J. Phys. , vol. 11, no. 10, pp. 1178-1193, Oct. 2013. https://www.researchgate.net/publication/257910110_Numerical_solutions_and_analysis_of_diffusion_for_new_generalized_fractional_Burgers_equation [27] S. Das and P. K. Gupta, "A mathematical model on fractional Lotka-Volterra equations, "J. Theor. Biol. , vol. 277, no. 1, pp. 1-6, May2011. http://www.sciencedirect.com/science/article/pii/S0022519311000609 [28] I. Grigorenko and E. Grigorenko, "Chaotic dynamics of the fractional Lorenz system, "Phys. Rev. Lett. , vol. 91, no. 3, pp. 034101, Jul. 2003. doi: 10.1103/PhysRevLett.91.034101 -
下载:
图(3)
计量
- 文章访问数: 1405
- HTML全文浏览量: 318
- PDF下载量: 636
- 被引次数: 0
