FAMILY OF INVARIANT CANTOR SETS AS ORBITS OF DIFFERENTIAL EQUATIONS II: JULIA SETS

Author:

CHEN YI-CHIUAN1,KAWAHIRA TOMOKI2,LI HUA-LUN3,YUAN JUAN-MING4

Affiliation:

1. Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan

2. Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan

3. Department of Applied Mathematics, Chung Hua University, Hsinchu 300, Taiwan

4. Department of Applied Mathematics, Providence University, Shalu, Taichung 433, Taiwan

Abstract

The Julia set of the quadratic map fμ(z) = μz(1 - z) for μ not belonging to the Mandelbrot set is hyperbolic, thus varies continuously. It follows that a continuous curve in the exterior of the Mandelbrot set induces a continuous family of Julia sets. The focus of this article is to show that this family can be obtained explicitly by solving the initial value problem of a system of infinitely coupled differential equations. A key point is that the required initial values can be obtained from the anti-integrable limit μ → ∞. The system of infinitely coupled differential equations reduces to a finitely coupled one if we are only concerned with some invariant finite subset of the Julia set. Therefore, it can be employed to find periodic orbits as well. We conduct numerical approximations to the Julia sets when parameter μ is located at the Misiurewicz points with external angle 1/2, 1/6, or 5/12. We approximate these Julia sets by their invariant finite subsets that are integrated along the reciprocal of corresponding external rays of the Mandelbrot set starting from the anti-integrable limit μ = ∞. When μ is at the Misiurewicz point of angle 1/128, a 98-period orbit of prescribed itinerary obtained by this method is presented, without having to find a root of a 298-degree polynomial. The Julia sets (or their subsets) obtained are independent of integral curves, but in order to make sure that the integral curves are contained in the exterior of the Mandelbrot set, we use the external rays of the Mandelbrot set as integral curves. Two ways of obtaining the external rays are discussed, one based on the series expansion (the Jungreis–Ewing–Schober algorithm), the other based on Newton's method (the OTIS algorithm). We establish tables comparing the values of some Misiurewicz points of small denominators obtained by these two algorithms with the theoretical values.

Publisher

World Scientific Pub Co Pte Lt

Subject

Applied Mathematics,Modelling and Simulation,Engineering (miscellaneous)

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1. From Cantor to semi-hyperbolic parameters along external rays;Transactions of the American Mathematical Society;2019-06-17

2. Simple proofs for the derivative estimates of the holomorphic motion near two boundary points of the Mandelbrot set;Journal of Mathematical Analysis and Applications;2019-05

3. New approximations for the area of the Mandelbrot set;Involve, a Journal of Mathematics;2017-03-07

4. Family of Smale–Williams Solenoid Attractors as Solutions of Differential Equations: Exact Formula and Conjugacy;International Journal of Bifurcation and Chaos;2015-09

5. Horseshoes, Entropy, Homoclinic Trajectories, and Lyapunov Stability;International Journal of Bifurcation and Chaos;2014-02

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