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Boundedness of Certain Forms of Jerky Dynamics

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Abstract

In this paper, we find regions in the bifurcation parameter space of certain forms of jerky dynamic systems where they are bounded. The method of analysis is based on a recent result about the boundedness of solutions of a certain type of third-order nonlinear differential equation with bounded delay. In particular, the boundedness of some chaotic attractors displayed by these systems is confirmed analytically.

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References

  1. Sprott J.C.: Some simple chaotic flows. Phys. Rev. E 50, R647–R650 (1994)

    Article  MathSciNet  Google Scholar 

  2. Sprott J.C.: Some simple chaotic jerk functions. Am. J. Phys. 65, 537–543 (1997)

    Article  Google Scholar 

  3. Sprott J.C.: Simplest dissipative chaotic flow. Phys. Lett. A228, 271–274 (1997)

    MathSciNet  Google Scholar 

  4. Moore D.W., Spiegel E.A.: A thermally excited nonlinear oscillator. Astrophys. J. 143, 871–887 (1966)

    Article  MathSciNet  Google Scholar 

  5. Tunc C.: Bound of solutions to third-order nonlinear differential equations with bounded delay. J. Frankl. Inst. 347, 415–425 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen G.: Controlling Chaos and Bifurcations in Engineering Systems. CRC Press, Boca Raton (1999)

    Google Scholar 

  7. Leonov G., Bunin A., Koksch N.: Attractor localization of the Lorenz system. Zeitschrift fur Angewandte Mathematik und Mechanik 67, 649–656 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Sun Y.: Solution bounds of generalized Lorenz chaotic systems. Chaos Solitons Fractals 40, 691–696 (2009)

    Article  MATH  Google Scholar 

  9. Li D., Lu J.A., Wu X., Chen G.: Estimating the bounds for the Lorenz family of chaotic systems. Chaos Solitons Fractals 23, 529–534 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zeraoulia E., Sprott J.C.: About the boundedness of 3D continuous-time quadratic systems. Nonlinear Oscillations 13(4), 1–8 (2010)

    MathSciNet  Google Scholar 

  11. Eichhorn R., Linz S.J., Hanggi P.: Transformations of nonlinear dynamical systems to jerky motion and its application to minimal chaotic flows. Phys. Rev. E 58, 7151–7164 (1998)

    Article  MathSciNet  Google Scholar 

  12. El’sgol’ts, L.E.: Introduction to the theory of differential equations with deviating arguments [translated from the Russian by Robert J. McLaughlin Holden-Day, Inc., SanFrancisco, California (1966)]

  13. Zhou T., Chen G.: Classification of chaos in 3-D autonomous quadratic systems-I Basic framework and methods. Int. J. Bifurc. Chaos 16, 2459–2479 (2006)

    Article  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, University of Tébessa, 12002, Tébessa, Algeria

    Zeraoulia Elhadj

  2. Department of Physics, University of Wisconsin, Madison, WI, 53706, USA

    J. C. Sprott

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  1. Zeraoulia Elhadj
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  2. J. C. Sprott
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Correspondence to Zeraoulia Elhadj.

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Elhadj, Z., Sprott, J.C. Boundedness of Certain Forms of Jerky Dynamics. Qual. Theory Dyn. Syst. 11, 199–213 (2012). https://doi.org/10.1007/s12346-011-0056-7

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  • DOI: https://doi.org/10.1007/s12346-011-0056-7

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