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International Journal of Technology Enhancements and Emerging Engineering Research (ISSN 2347-4289)
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International Journal of Technology Enhancements and Emerging Engineering Research  
International Journal of Technology Enhancements and Emerging Engineering Research

Website: http://www.ijteee.org

ISSN 2347-4289



Impending Role Of Period Three Points In Chaotic Functions Through Sturm Method And Matlab Programming

[Full Text]

 

AUTHOR(S)

R. S. Thakkar, P. J. Bhatt

 

KEYWORDS

Keywords : Chaotic Behaviour of a function, Bifurcation Diagram, Sturm Method, Period Three points, MATLAB programming

 

ABSTRACT

ABSTRACT: Chaos Theory and Dynamical Systems has been considered as one of the most significant breakthroughs in Mathematics in this century. Its applications in a wide range of subjects including Physics, Biology, Chemistry, Ecology, Fluid Mechanics, Engineering, Economics etc., have made the field very attractive and important for the researchers from various disciplines. The salient feature of the chaos theory is that even simplest looking maps illustrate virtually every important phenomena that occur in the Dynamical Systems. The logistic map f (x) = k x ( 1 – x ) is a popular example of such a behaviour. This family has been one of the simplest of the nonlinear maps, but it exhibits the various aspects of dynamical systems varying from stability to chaos. However this concept needs to be generalized. The quite unknown dynamics of a cubic family of functions defined by f ( x ) = x3 + x shows some interesting similarities as well as differences with this family of functions. The advent of modern computers has quickened the pace of development in the field of Chaos. Matlab programming can also play a vital role to visualize the inner properties of chaotic functions, including one of the three essential properties of chaotic functions viz. existence of period three points. This paper explains the existence of period three points for cubic family of functions through the Sturm method for finding the solution of the higher degree polynomials and Matlab programming.

 

REFERENCES

[1] Devaney R. L., An introduction to Chaotic Dynamical Systems, Second Edition, Addition Wesley, Redwood city, 1989

[2] Michel Vellekoop ; Raoul Berglund , On intervals, Transitivity = Chaos, American Mathematical Monthly, Volume 101, Issue 4 ( April 1994), 353-355

[3] Kathleen T. Alligood, Tim D. Sauer, James A. Yorke , CHAOS – An introduction to dynamical systems, Springer Verlag, New York Inc., 1997

[4] Steven H. Strogatz , Nonlinear Dynamics and chaos : With applications to Physics, Biology, Chemistry and Engineering.

[5] Tien-Yien Li and James A. Yorke, ‘Period three implies Chaos’ American Mathematical Monthly, 1975.

[6] E. J. Barbeau , Polynomials, Page 164
[7] Brian R. Hunt, Ronald Lipsman, Jonathan M. Rosen-berg : A guide to MATLAB for Beginners and Experienced Users, Cambridge University Press, 2001