Research Article
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Discrete Superior Hyperbolicity in Chaotic Maps

Year 2021, Volume: 3 Issue: 1, 34 - 42, 30.06.2021
https://doi.org/10.51537/chaos.936679

Abstract

In the last few decades, the dynamics of one-dimensional chaotic maps have gained the tremendous attention of scientists and scholars due to their remarkable properties such as period-doubling, chaotic evolution, Lyapunov exponent, etc. The term hyperbolicity, another important property of chaotic maps is used to examine the regular and irregular behavior of the dynamical systems. In this article, we deal with the hyperbolicity and stabilization of fixed states using a superior two-step feedback system. Due to the superiority in the chaotic evolution of one-dimensional maps in the superior system we are encouraged to examine the hyperbolicity and stabilization in chaotic maps. The hyperbolic notion, hyperbolicity in periodic states of prime order, stabilization, and the hyperbolic set of the chaotic maps are studied. The numerical, as well as experimental simulations, are carried out, followed by theorems, examples, remarks, functional plots, and bifurcation diagrams.

Supporting Institution

King Abdulaziz University, Jeddah, Saudi Arabia

Project Number

FP-108-42

Thanks

Thanks.

References

  • Adiyaman, Y., S. Emiroglu, M. Ucar and M. Yildiz, 2020 Dynamical analysis, electronic circuit design and control application of a different chaotic system, Chaos Theory and Applications 02: 10-16.
  • Akgul, A., Kaçar, S., Aricıoglu, B., and Pehlivan, I., Text encryption by using one-dimensional chaos generators and nonlinear equations. In 2013 8th International Conference on Electrical and Electronics Engineering (ELECO), IEEE 320-323.
  • Alligood, K. T., T. D. Sauer and J. A. Yorke, 1996 Chaos : An Introduction to Dynamical Systems, Springer Verlag, New York Inc.
  • Andrecut, M., 1998 Logistic map as a random number generator, International Journal of Modern Physics B 12: 101-102.
  • Ashish, J. Cao and R. Chugh, 2018 Chaotic behavior of logistic map in superior orbit and an improved chaos-based traffic control model, Nonlinear Dynamics 94: 959-975.
  • Ashish and J. Cao, 2019a A novel fixed point feedback approach studying the dynamcial behaviour of standard logistic map, International Journal of Bifurcation and Chaos 29: 1950010-16, 16 pages.
  • Ashish, J. Cao and R. Chugh, 2019b Controlling chaos using superior feedback technique with applications in discrete traffic models, International Journal of Fuzzy System 21: 1467-1479.
  • Ashish, J. Cao and R. Chugh, 2021 Discrete chaotification in modulated logistic system, International Journal of Bifurcation and Chaos 31: 2150065, 14 Pages.
  • Aulbach, B. and B. Kieninger, 2004 An elementary proof for hyperbolicity and chaos of the logistic maps, Journal of Difference Equations and Applications 10: 1243-1250.
  • Ausloos, M. and M. Dirickx, 2006 The Logistic Map and the Route to Chaos : from the Beginnings to Modern Applications, Springer Verlag, New York Inc.
  • Baptista, M. S., 1998 Logistic map as a random number generator, Physics Letter A 240: 50-54.
  • Chugh, R., M. Rani and Ashish, 2012 Logistic map in Noor orbit, Chaos and Complexity Letter 6, 167-175.
  • Devaney, R. L., 1948 An Introduction to Chaotic Dynamical Systems, 2nd Edition (Addison-Wesley).
  • Devaney, R. L., 1992 A First Course in Chaotic Dynamical Systems: Theory and Experiment, (Addison-Wesley).
  • Glendinning, P., 2001 Hyperbolicity of the invariant set for the logistic map with m > 4, Nonlinear Analysis 47: 3323-3332.
  • Guckenheimer, J., 1979 Sensitive dependence to initial conditions for one-dimensional maps, Communications in Mathematical Physics 70: 133-160.
  • Holmgren, R. A., 1994 A First Course in Discrete Dynamical Systems, Springer Verlag, New York Inc.
  • Jonassen, T. M., 2002 On the Concept of Hyperbolicity, Oslo Univ. College Report Series 21:, ISBN 82-579-4155-7.
  • Kraft, R. L., 1999 Chaos, cantor sets and hyperbolicity for the logistic maps, Transactions of the American Mathematical Society 106: 400-408.
  • Kumar, V., Khamosh and Ashish, 2020 An empirical approach to study the stability og generalized logistic map in superior orbit, Advances In Mathematics: Scientific Journal 10: 2094-2109.
  • Mann,W. R., 1953 Mean value methods in iteration, Proceedings of American Mathematical Society 04: 506-510.
  • Martelli, M., 1999 Chaos : An Introduction to Discrete Dynamical Systems and Chaos,Wiley-Interscience Publication, New York Inc.
  • Melo, W. de. and S. J. van Strien, 1993 One-dimensional dynamics, Springer, Berlin.
  • Misiurewicz, M., 1976 Absolutely continuous measures for certain maps of an interval, Publications Mathematiques de l’Institut des Hautes Etudes Scientifiques, 261: 459-475.
  • Newhouse, S. J., 1981 The abundance of wild hyperbolic set and non-smooth stable sets for diffeomorphism, Publications Mathematiques de l’Institut des Hautes Etudes Scientifiques, 53: 17-51.
  • Poincare, H., 1899 Les Methods Nouvells de la Mecanique Leleste, Gauthier Villars, Paris.
  • Robinson, C., 1995 Dynamical Systems: Stabilily, Symbolic Dynamics, and Chaos, CRC Press.
  • Saha, L. M., L. Bharti and R. K. Mohanty, 2010 Study of bifurcation and hyperbolicity in discrete dynamical systems, Iranian Journal of Science and Technology, 34: 1-12.
  • Saha, L. M., R. K. Mohanty and L. Bharti, 2009 Hyperbolicity and chaos in discrete systems, International Journal of Applied Mathematics and Mechanics, 05: 48-56.
  • Sharkovsky, A. N., Y. L. Maistrenko and E. Y. Romanenko, 1993 Difference Equations and Their Applications, Kluwer Academic Publisher.
  • Volos, C. K., Akgul, A., Pham, V. T., and Baptista, M. S., 2018 Antimonotonicity, crisis and multiple attractors in a simple memristive circuit. Journal of Circuits, Systems and Computers, 27(02): 1850026.
  • Wang, X., Akgul, A., Cicek, S., Pham, V. T., and Hoang, D. V., 2017 A chaotic system with two stable equilibrium points: Dynamics, circuit realization and communication application. International Journal of Bifurcation and Chaos, 27(08): 1750130.
Year 2021, Volume: 3 Issue: 1, 34 - 42, 30.06.2021
https://doi.org/10.51537/chaos.936679

Abstract

Project Number

FP-108-42

References

  • Adiyaman, Y., S. Emiroglu, M. Ucar and M. Yildiz, 2020 Dynamical analysis, electronic circuit design and control application of a different chaotic system, Chaos Theory and Applications 02: 10-16.
  • Akgul, A., Kaçar, S., Aricıoglu, B., and Pehlivan, I., Text encryption by using one-dimensional chaos generators and nonlinear equations. In 2013 8th International Conference on Electrical and Electronics Engineering (ELECO), IEEE 320-323.
  • Alligood, K. T., T. D. Sauer and J. A. Yorke, 1996 Chaos : An Introduction to Dynamical Systems, Springer Verlag, New York Inc.
  • Andrecut, M., 1998 Logistic map as a random number generator, International Journal of Modern Physics B 12: 101-102.
  • Ashish, J. Cao and R. Chugh, 2018 Chaotic behavior of logistic map in superior orbit and an improved chaos-based traffic control model, Nonlinear Dynamics 94: 959-975.
  • Ashish and J. Cao, 2019a A novel fixed point feedback approach studying the dynamcial behaviour of standard logistic map, International Journal of Bifurcation and Chaos 29: 1950010-16, 16 pages.
  • Ashish, J. Cao and R. Chugh, 2019b Controlling chaos using superior feedback technique with applications in discrete traffic models, International Journal of Fuzzy System 21: 1467-1479.
  • Ashish, J. Cao and R. Chugh, 2021 Discrete chaotification in modulated logistic system, International Journal of Bifurcation and Chaos 31: 2150065, 14 Pages.
  • Aulbach, B. and B. Kieninger, 2004 An elementary proof for hyperbolicity and chaos of the logistic maps, Journal of Difference Equations and Applications 10: 1243-1250.
  • Ausloos, M. and M. Dirickx, 2006 The Logistic Map and the Route to Chaos : from the Beginnings to Modern Applications, Springer Verlag, New York Inc.
  • Baptista, M. S., 1998 Logistic map as a random number generator, Physics Letter A 240: 50-54.
  • Chugh, R., M. Rani and Ashish, 2012 Logistic map in Noor orbit, Chaos and Complexity Letter 6, 167-175.
  • Devaney, R. L., 1948 An Introduction to Chaotic Dynamical Systems, 2nd Edition (Addison-Wesley).
  • Devaney, R. L., 1992 A First Course in Chaotic Dynamical Systems: Theory and Experiment, (Addison-Wesley).
  • Glendinning, P., 2001 Hyperbolicity of the invariant set for the logistic map with m > 4, Nonlinear Analysis 47: 3323-3332.
  • Guckenheimer, J., 1979 Sensitive dependence to initial conditions for one-dimensional maps, Communications in Mathematical Physics 70: 133-160.
  • Holmgren, R. A., 1994 A First Course in Discrete Dynamical Systems, Springer Verlag, New York Inc.
  • Jonassen, T. M., 2002 On the Concept of Hyperbolicity, Oslo Univ. College Report Series 21:, ISBN 82-579-4155-7.
  • Kraft, R. L., 1999 Chaos, cantor sets and hyperbolicity for the logistic maps, Transactions of the American Mathematical Society 106: 400-408.
  • Kumar, V., Khamosh and Ashish, 2020 An empirical approach to study the stability og generalized logistic map in superior orbit, Advances In Mathematics: Scientific Journal 10: 2094-2109.
  • Mann,W. R., 1953 Mean value methods in iteration, Proceedings of American Mathematical Society 04: 506-510.
  • Martelli, M., 1999 Chaos : An Introduction to Discrete Dynamical Systems and Chaos,Wiley-Interscience Publication, New York Inc.
  • Melo, W. de. and S. J. van Strien, 1993 One-dimensional dynamics, Springer, Berlin.
  • Misiurewicz, M., 1976 Absolutely continuous measures for certain maps of an interval, Publications Mathematiques de l’Institut des Hautes Etudes Scientifiques, 261: 459-475.
  • Newhouse, S. J., 1981 The abundance of wild hyperbolic set and non-smooth stable sets for diffeomorphism, Publications Mathematiques de l’Institut des Hautes Etudes Scientifiques, 53: 17-51.
  • Poincare, H., 1899 Les Methods Nouvells de la Mecanique Leleste, Gauthier Villars, Paris.
  • Robinson, C., 1995 Dynamical Systems: Stabilily, Symbolic Dynamics, and Chaos, CRC Press.
  • Saha, L. M., L. Bharti and R. K. Mohanty, 2010 Study of bifurcation and hyperbolicity in discrete dynamical systems, Iranian Journal of Science and Technology, 34: 1-12.
  • Saha, L. M., R. K. Mohanty and L. Bharti, 2009 Hyperbolicity and chaos in discrete systems, International Journal of Applied Mathematics and Mechanics, 05: 48-56.
  • Sharkovsky, A. N., Y. L. Maistrenko and E. Y. Romanenko, 1993 Difference Equations and Their Applications, Kluwer Academic Publisher.
  • Volos, C. K., Akgul, A., Pham, V. T., and Baptista, M. S., 2018 Antimonotonicity, crisis and multiple attractors in a simple memristive circuit. Journal of Circuits, Systems and Computers, 27(02): 1850026.
  • Wang, X., Akgul, A., Cicek, S., Pham, V. T., and Hoang, D. V., 2017 A chaotic system with two stable equilibrium points: Dynamics, circuit realization and communication application. International Journal of Bifurcation and Chaos, 27(08): 1750130.
There are 32 citations in total.

Details

Primary Language English
Subjects Software Engineering (Other)
Journal Section Research Articles
Authors

Ashish Ashish 0000-0001-9598-3393

Jinde Cao 0000-0003-3133-7119

Fawaz Alsaadi This is me 0000-0003-0041-3158

A. K. Malik This is me 0000-0002-1520-0115

Project Number FP-108-42
Publication Date June 30, 2021
Published in Issue Year 2021 Volume: 3 Issue: 1

Cite

APA Ashish, A., Cao, J., Alsaadi, F., Malik, A. K. (2021). Discrete Superior Hyperbolicity in Chaotic Maps. Chaos Theory and Applications, 3(1), 34-42. https://doi.org/10.51537/chaos.936679

Chaos Theory and Applications in Applied Sciences and Engineering: An interdisciplinary journal of nonlinear science 23830 28903   

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