Research Papers

On a Discrete Chaos Induction Via an Aperiodic Kicks Pattern

[+] Author and Article Information
Mehdi Nategh

Department of Mathematics and Statistics,
Missouri University of Science and Technology,
Rolla MO, 65401
e-mail: nateghm@mst.edu

Dumitru Baleanu

Department of Mathematics,
Çankaya University,
Ankara 06790, Turkey
e-mail: dumitru@cankaya.edu.tr

Mohammad Reza Valinejad

Arsh Electronic Company,
Babol, Iran
e-mail: mrv.ir152@gmail.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 10, 2016; final manuscript received October 7, 2016; published online January 20, 2017. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 12(4), 041008 (Jan 20, 2017) (6 pages) Paper No: CND-16-1378; doi: 10.1115/1.4035078 History: Received August 10, 2016; Revised October 07, 2016

In this work, a class of kicked systems perturbed with an irregular kicks pattern is studied and formation of a chaos in the senses of Devaney and Li–Yorke in the corresponding discretized system is investigated. Beside a discussion on chaotic stability, an example is presented. Then, the existence of a period three orbit of a 2D map which governs a class of dynamic problems on time scales is studied. As an application, a chaotic encryption scheme for a time-dependent plain text with the help of chaos induction in the sense of Li–Yorke is presented.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Aulbach, B. , and Kieninger, B. , 2001, “ On Three Definitions of Chaos,” Nonlinear Dyn. Syst. Theory, 1(1), pp. 23–37.
Morel, C. , Vlad, R. , and Morel, J.-Y. , 2008, “ Anticontrol of Chaos Reduces Spectral Emissions,” ASME J. Comput. Nonlinear Dyn., 3(4), p. 041009. [CrossRef]
Razminia, A. , and Baleanu, D. , 2013, “ Fractional Hyperchaotic Telecommunication Systems: A New Paradigm,” ASME J. Comput. Nonlinear Dyn., 8(3), p. 031012. [CrossRef]
Feng, C. , Cai, L. , Kang, Q. , Wang, S. , and Zhang, H. , 2015, “ Novel Hyperchaotic System and Its Circuit Implementation,” ASME J. Comput. Nonlinear Dyn., 10(6), p. 061012. [CrossRef]
Behnia, S. , Akhshani, A. , Ahadpour, S. , Mahmodi, H. , and Akhavan, A. , 2007, “ A Fast Chaotic Encryption Scheme Based on Piecewise Nonlinear Chaotic Maps,” Phys. Lett. A, 366(4–5), pp. 391–396. [CrossRef]
Zaslavsky, G. M. , 2005, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, UK.
Abdullaev, S. S. , 2006, Construction of Mappings for Hamiltonian Systems and Their Applications, Springer, Berlin.
Baptista, M. O. S. , and Clades, I. L. , 1996, “ Dynamics of the Kicked Logistic Map,” Chaos, Solitons Fractals, 7(3), pp. 326–336.
Schuster, H. G. , and Just, W. , 2005, Deterministic Chaos, An Introduction, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany.
Zheng, Y. , and Kobe, D. H. , 2005, “ Numerical Solution of Classical Kicked Rotor and Local Lyapunov Exponents,” Phys. Lett. A, 334(4), pp. 306–311. [CrossRef]
Zheng, Y. , and Kobe, D. H. , 2006, “ Anomalous Momentum Diffusion in the Classical Kicked Rotor,” Chaos, Solitons Fractals, 28(2), pp. 395–402. [CrossRef]
Edelman, M. , and Tarasov, V. E. , 2009, “ Fractional Standard Map,” Phys. Lett. A, 374(2), pp. 279–285. [CrossRef]
Wu, G. C. , Baleanu, D. , and Zeng, S. D. , 2014, “ Discrete Chaos in Fractional Sine and Standard Maps,” Phys. Lett. A, 378(5–6), pp. 484–487. [CrossRef]
Agarwal, R. , Bohner, M. , O'Regan, D. , and Peterson, A. , 2002, “ Dynamic Equations on Time Scales: A Survey,” J. Comput. Appl. Math., 141(1–2), pp. 1–26. [CrossRef]
Bohner, M. , and Peterson, A. , 2001, Dynamic Equations on Time Scales: An Introduction With Applications, Springer, Heidelberg, Germany.
Tarasov, V. E. , 2010, Fractional Dynamics, Applications of Fractional Calculus to Dynamics of Particles Fields and Media, Springer Verlag, Heidelberg, Germany.
Devaney, R. L. , 1989, An Introduction to Chaotic Dynamical Systems, Addison Wesley Publication, Boston, MA.
Banks, J. , Brooks, J. , Cairns, G. , Davis, G. , and Stacey, R. , 1992, “ On Devaney's Definition of Chaos,” Am. Math. Mon., 99(4), pp. 332–334. [CrossRef]
Li, T. Y. , and Yorke, J. A. , 1975, “ Period Three Implies Chaos,” Am. Math. Mon., 82(10), pp. 985–992. [CrossRef]
Huang, W. , and Ye, X. , 2002, “ Devaneys Chaos or 2-Scattering Implies LiYorkes Chaos,” Topol. Its Appl., 117(3), pp. 259–272. [CrossRef]
Li, J. , and Ye, X. , 2016, “ Recent Development of Chaos Theory in Topological Dynamics,” Acta. Math. Sin.-English Ser., 32(1), pp. 83–114.
Shi, Y. , and Chen, G. , 2004, “ Chaos of Discrete Dynamical Systems in Complete Metric Spaces,” Chaos, Solitons Fractals, 22(3), pp. 555–571. [CrossRef]
Shi, Y. , and Chen, G. , 2005, “ Discrete Chaos in Banach Spaces,” Sci. China, Ser. A: Math., 48(2), pp. 222–238. [CrossRef]
Ostrowski, A. , 1938, “ Ü ber die Absolutabweichung einer differentiebaren Funcktion von ihrem Integralmittelwert,” Commentarii Mathematici Helvetici, 10(1937–1938), pp. 226–227. [CrossRef]


Grahic Jump Location
Fig. 1

Projected orbits in X–P plane with initial values x0=0.1, p0=0.3, μ0=0.9999 and x0=0.1, p0=0.3, μ0=0.999 represented by scattered points and vertical line x ∼ 0.3 respectively

Grahic Jump Location
Fig. 2

A schematic view for mapping a block (xk,μk) to the encrypted block Fm(xk,μk) via introducing a new time scale Tk with its corresponding Pk



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In