Verification of Feigenbaum’s Universality in One Dimensional Nonlinear Mathematical Model

Anil Kumar Jain

doi:https://doi.org/10.14445/22315373/IJMTT-V13P511

International Journal of Mathematics Trends and Technology

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Volume 13 | Number 2 | Year 2014 | Article Id. IJMTT-V13P511 | DOI : https://doi.org/10.14445/22315373/IJMTT-V13P511

Verification of Feigenbaum’s Universality in One Dimensional Nonlinear Mathematical Model

Anil Kumar Jain

Citation :

Anil Kumar Jain, "Verification of Feigenbaum’s Universality in One Dimensional Nonlinear Mathematical Model," International Journal of Mathematics Trends and Technology (IJMTT), vol. 13, no. 2, pp. 79-89, 2014. Crossref, https://doi.org/10.14445/22315373/IJMTT-V13P511

Abstract

Here we consider a rational nonlinear mathematical model f(x)=(μ x)/(1+(a x)b ) with μ as control parameter and a,b are taken as constants and examine period-doubling route to chaos . We observe that the route followed by this map is universal in the sense of Feigenbaum’s universality constant. In order to verify an universal route from order to chaos through period doubling bifurcations appropriate numerical methods such as Newton-Raphson method, Bisection method etc. are used to obtain periodic points and bifurcation points of different period 20 ,21 ,22 ,23,24……and find Feigenbaum Universal Constant(δ)=4.66920161029… with the help of bifurcation points calculated numerically. Also with the help of experimental bifurcation points and Feigenbaum delta, the accumulation point is calculated numerically .The period doubling scenario explains us how the behaviour of the map changes from regularity to chaotic one. Then periodic behaviours of the map are established by plotting the Time-series graphs. Lastly chaotic region has also been confirmed by obtaining positive Lyapunov Exponents at some parametric values.

Keywords

Doubling Bifurcation, Periodic points, Feigenbaum Universal Constant, Time Series, Lyapunov Exponent

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