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eISSN:2278-5299

International Journal of Latest Research in Science and Technology

DOI:10.29111/ijlrst   ISRA Impact Factor:3.35,  Peer-reviewed, Open-access Journal

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Analysis of the Chaotic Nature in a Cubic Map

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International Journal of Latest Research in Science and Technology Vol.8 Issue 6, pp 20-27,Year 2019

ANALYSIS OF THE CHAOTIC NATURE IN A CUBIC MAP

Pramila Kumari Prajapati , Premchand Sharma , Sankar Haloi

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Received : 23 November 2019; Accepted : 29 December 2019 ; Published : 31 December 2019

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Article No. 10982
Abstract

In this paper we consider a cubic map ax3 + bx2 + cx + d = 0, where a,b,c and d are parameters. A cubic map is a bimodal map which is an iterated discrete-time dynamical system exhibiting chaotic behavior having infinitely many unstable periodic points. Here we have proved the following: 1) This map exhibits transition to period doubling route and finally leads to a bounded chaotic region that possesses many exciting dynamical properties to be studied . 2) Its periodic doubling bifurcation and the reverse bifurcation are studied for different values of the parameters. 3) An analysis has been carried out by using Lyapunov Exponent and Time Series technique . 4)Feigenbaum Universal Delta along with accumulation point have been calculated in order to establish Feigenbaum universal Properties . 5) Many open problems have been posed

Key Words   
Period doubling Bifurcation, Reverse bifurcation, Lyapunov Exponent, Feigenbaum Delta, Chaos,

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References
  1. Arrowsmith, D.K and Place, C.M., An Introduction to Dynamical System, Cambridge University Press,(1994).
  2. Das, N., Sharmah, R. and Dutta, N., Period doubling bifurcation  and  associated  universal  properties in the Verhulst population model, International J. of Math. Sci. & Engg. Appls., Vol. 4 No. I (March 2010), pp. 1-14
  3. Devaney, R.L.,An introduction to Chaotic Dynamical Systems, 2nd ed, Addison Wesley(1989)
  4. Denny, G., Encounters with chaos, McGraw-Hill,Inc.(1992)
  5. Dutta, T.K. & Prajapati, P.K., Analysis On Stability Of Periodic  Points,  Period  Doubling Bifurcation And Lypunov Exponent In A Chaotic Model., International Research Journal Of Mathematics, Engineering & IT, ISSN: (2349-0322), Vol-1, Issue-8 (December 2014), pp.1-12
  6. Feigenbaum Mitchell, Quantitative Universality for a class of Nonlinear Transformations, Journal of Statistical Physics, 19(1), pp25-52, July 1978.
  7. Hirsch, M.W. , Stability and Convergence in strong monotone Dynamical Systems, J. reine angew. Math., 383 (1988), 53-65.
  8. Kuznetsov A.P., Kuzneosov S P., Salaev I.R. and Chua L O , Multi-parameter critically in Chua’s circuit at Period- Doubling Transition to Chaos, International Journal  of Bifurcation and Chaos, Vol 06, No 1, pp119-148, 1996.
  9. Kuznetsov A.P., Turukina L.V. and Mosekilde E., Dynamical Systems of Different Classes of Models of Kicked Nonlinear Oscillator, International Journal  of Bifurcation and Chaos, Vol 11, No 04, pp 1065- 1077, 2011.
  10. Sarmah, H. & Das, M., Various Bifurcation in a Cubic Map, International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954, Issue 4 volume 3, May-June 2014,827-846
  11. Smith Reginald D., Period Doubling, Information Entropy and Estimate of Feigenbaum Constants, International Journal  of Bifurcation and Chaos, Vol 23, No 11, 1350190 (2013).
To cite this article

Pramila Kumari Prajapati , Premchand Sharma , Sankar Haloi , " Analysis Of The Chaotic Nature In A Cubic Map ", International Journal of Latest Research in Science and Technology . Vol. 8, Issue 6, pp 20-27 , 2019

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