Several Chaotic Approaches of One Dimensional Doubling Map
Keywords:
Approaches, Orbit, Sensitivity, Staircase Diagram, Trajectories, Transitivity.Abstract
In this paper, we study basic dynamical behavior of one-dimensional Doubling map. Especially emphasis is given on the chaotic behaviors of the said map. Several approaches of chaotic behaviors by some pioneers it is found that the Doubling map is chaotic in different senses. We mainly focused on Orbit Analysis, Sensitivity to Initial Conditions, Sensitivity to Numerical Inaccuracies, Trajectories and Staircase Diagram of the Doubling map. The graphical representations show that this map is chaotic in different senses. The behavior of the said map is found irregular, that is, chaotic.
References
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[2] Dynamical Systems and Ergodic Theory-The Doubling Map, Corinna Ulcigrai, University of Bristol.
[3] Evan Dummit, 2015, v. 1.00, Dynamics, Chaos, and Fractals (part 3): Chaotic Dynamics.
[4] Evgeny Demidov, Chaotic 1 D maps.
[5] Heikki Ruskeepaa, Mathematica Navigator, 3rd Edition.
[6] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney’s Definition of chaos, American Mathematic Monthly 99 (1992), 332-334
[7] Li, T.-y., and Yorke, J., “Period Three Implies Chaos.” American Mathematical Monthly 82 (1975), 985-992.
[8] P. Ahmed, Chaotic homeomorphisms in one-dimensional dynamics, Proc., Iwate Univ., Japan, May 5-6, 2001.
[9] R. L. Devaney, A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addision-Wesley, New York,1992.
[10] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Second Edition, Addision-Wesley, RedwoodCity,1989.
[11] R. L. Devaney and L. Keen, Chaos and Fractals: The Mathematics Behind the Computer Graphics, American Mathematical Society, Providence, 1989.
[12] S. Kolyada and L. Snoha, Some aspects of topological transitivity-A survey, 1997.
[13] S. Wiggins, Chaotic Transport in Dynamical Systems, Interdisciplinary Applied Mathematics Series, Volume 2, Spring-Verlag, Berlin, 1992.
[14] Wolf, A. “Quantifying Chaos with Lyapunov exponents,” in Chaos, edited by A. V. Holden, Princeton University Press, 1986.
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Published
2018-05-27
How to Cite
Hosen, M. Z., Zillu, M. M., & Ahmed, P. (2018). Several Chaotic Approaches of One Dimensional Doubling Map. American Scientific Research Journal for Engineering, Technology, and Sciences, 43(1), 110–133. Retrieved from https://asrjetsjournal.org/index.php/American_Scientific_Journal/article/view/3703
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