Discrete Chaos in Fractional Henon Map
Author(s)
Tongchun Hu*
ABSTRACT
In this study, a discrete fractional Henon map is proposed in the Caputo discrete delta’s sense. The results show that the discrete fractional calculus is an efficient tool and the maps derived in this way have simpler forms but hold rich dynamical behaviors.
Cite this paper
Hu, T. (2014) Discrete Chaos in Fractional Henon Map. Applied Mathematics, 5, 2243-2248. doi: 10.4236/am.2014.515218.
Hu, T. (2014) Discrete Chaos in Fractional Henon Map. Applied Mathematics, 5, 2243-2248. doi: 10.4236/am.2014.515218.
References
[1] Hartley, T.T. and Lorenzo, C.F. (2002) Dynamics and Control of Initialized Fractional-Order Systems. Nonlinear Dynamics, 29, 201-233. http://dx.doi.org/10.1023/A:1016534921583 [2] Hwang, C., Leu, J.F. and Tsay, S.Y. (2002) A Note on Time-Domain Simulation of Feedback Fractional-Order Systems. IEEE Transactions on Automatic Control, 47, 625-631.
http://dx.doi.org/10.1109/9.995039 [3] Podlubny, I., Petras, I., Vinagre, B.M., O’Leary, P. and Dorcak, L. (2002) Analogue Realizations of Fractional-Order Controllers. Nonlinear Dynamics, 29, 281-296.
http://dx.doi.org/10.1023/A:1016556604320 [4] Babakhani, A., Baleanu, D. and Khanbabaie, R. (2012) Hopf Bifurcation for a Class of Fractional Differential Equations with Delay. Nonlinear Dynamics, 29, 721-729.
http://dx.doi.org/10.1007/s11071-011-0299-5 [5] Baleanu, D., Diethelm, K., Scalas, E. and Trujillo, J.J. (2012) Fractional Calculus Models and Numerical Methods. World Scientific, Boston. [6] Li, C.P. and Peng, P.J. (2004) Chaos in Chen’s System with a Fractional Order. Chaos Solitons & Fractals, 22, 443-450. http://dx.doi.org/10.1016/j.chaos.2004.02.013 [7] Atici, F.M. and Eloe, P.W. (2009) Initial Value Problems in Discrete Fractional Calculus. Proceedings of the American Mathematical Society, 137, 981-989.
http://dx.doi.org/10.1090/S0002-9939-08-09626-3 [8] Abdeljawad, T. (2011) On Riemann and Caputo Fractional Differences. Computers & Mathematics with Applications, 62, 1602-1611.
http://dx.doi.org/10.1016/j.camwa.2011.03.036 [9] Chen, F.L., Luo, X.N. and Zhou, Y. (2011) Existence Results for Nonlinear Fractional Difference Equation. Advances in Difference Equations, 2011, Article ID: 713201.
http://dx.doi.org/10.1155/2011/713201 [10] Xiao, H., Ma, Y. and Li, C.P. (2014) Chaotic Vibration in Fractional Maps. Journal of Vibration and Control, 20, 964-972. http://dx.doi.org/10.1177/1077546312473769 [11] Wu, G.C., Baleanu, D. and Zeng, S.D. (2014) Discrete Chaos in Fractional Sine and Standard Maps. Physics Letters A, 378, 484-487. http://dx.doi.org/10.1016/j.physleta.2013.12.010 [12] Wu, G.C. and Baleanu, D. (2014) Discrete Chaos in Fractional Delayed Logistic Maps. Nonlinear Dynamics, in Press. [13] Tarasov, V.E. and Edelman, M. (2010) Fractional Dissipative Standard Map. Chaos, 20, Article ID: 02327.
http://dx.doi.org/10.1063/1.3443235
[1] Hartley, T.T. and Lorenzo, C.F. (2002) Dynamics and Control of Initialized Fractional-Order Systems. Nonlinear Dynamics, 29, 201-233. http://dx.doi.org/10.1023/A:1016534921583 [2] Hwang, C., Leu, J.F. and Tsay, S.Y. (2002) A Note on Time-Domain Simulation of Feedback Fractional-Order Systems. IEEE Transactions on Automatic Control, 47, 625-631.
http://dx.doi.org/10.1109/9.995039 [3] Podlubny, I., Petras, I., Vinagre, B.M., O’Leary, P. and Dorcak, L. (2002) Analogue Realizations of Fractional-Order Controllers. Nonlinear Dynamics, 29, 281-296.
http://dx.doi.org/10.1023/A:1016556604320 [4] Babakhani, A., Baleanu, D. and Khanbabaie, R. (2012) Hopf Bifurcation for a Class of Fractional Differential Equations with Delay. Nonlinear Dynamics, 29, 721-729.
http://dx.doi.org/10.1007/s11071-011-0299-5 [5] Baleanu, D., Diethelm, K., Scalas, E. and Trujillo, J.J. (2012) Fractional Calculus Models and Numerical Methods. World Scientific, Boston. [6] Li, C.P. and Peng, P.J. (2004) Chaos in Chen’s System with a Fractional Order. Chaos Solitons & Fractals, 22, 443-450. http://dx.doi.org/10.1016/j.chaos.2004.02.013 [7] Atici, F.M. and Eloe, P.W. (2009) Initial Value Problems in Discrete Fractional Calculus. Proceedings of the American Mathematical Society, 137, 981-989.
http://dx.doi.org/10.1090/S0002-9939-08-09626-3 [8] Abdeljawad, T. (2011) On Riemann and Caputo Fractional Differences. Computers & Mathematics with Applications, 62, 1602-1611.
http://dx.doi.org/10.1016/j.camwa.2011.03.036 [9] Chen, F.L., Luo, X.N. and Zhou, Y. (2011) Existence Results for Nonlinear Fractional Difference Equation. Advances in Difference Equations, 2011, Article ID: 713201.
http://dx.doi.org/10.1155/2011/713201 [10] Xiao, H., Ma, Y. and Li, C.P. (2014) Chaotic Vibration in Fractional Maps. Journal of Vibration and Control, 20, 964-972. http://dx.doi.org/10.1177/1077546312473769 [11] Wu, G.C., Baleanu, D. and Zeng, S.D. (2014) Discrete Chaos in Fractional Sine and Standard Maps. Physics Letters A, 378, 484-487. http://dx.doi.org/10.1016/j.physleta.2013.12.010 [12] Wu, G.C. and Baleanu, D. (2014) Discrete Chaos in Fractional Delayed Logistic Maps. Nonlinear Dynamics, in Press. [13] Tarasov, V.E. and Edelman, M. (2010) Fractional Dissipative Standard Map. Chaos, 20, Article ID: 02327.
http://dx.doi.org/10.1063/1.3443235