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Unveiling the fractal structure of Julia sets with Lagrangian descriptors
- Source :
- Communications in Nonlinear Science and Numerical Simulation. 91:105417
- Publication Year :
- 2020
- Publisher :
- Elsevier BV, 2020.
-
Abstract
- In this paper we explore by means of the method of Lagrangian descriptors the Julia sets arising from complex maps, and we analyze their underlying dynamics. In particular, we take a look at two classical examples: the quadratic mapping $z_{n+1} = z^2_n + c$, and the maps generated by applying Newton's method to find the roots of complex polynomials. To achieve this goal, we provide an extension of this scalar diagnostic that is capable of revealing the phase space of open maps in the complex plane, allowing us to avoid potential issues of orbits escaping to infinity at an increasing rate. The simple idea is to compute the p-norm version of Lagrangian descriptors, not for the points on the complex plane, but for their projections on the Riemann sphere in the extended complex plane. We demonstrate with several examples that this technique successfully reveals the rich and intricate dynamical features of Julia sets and their fractal structure.<br />Comment: 12 pages, 7 figures. Submitted
- Subjects :
- Numerical Analysis
Pure mathematics
Applied Mathematics
Scalar (mathematics)
FOS: Physical sciences
Riemann sphere
Dynamical Systems (math.DS)
Nonlinear Sciences - Chaotic Dynamics
01 natural sciences
Julia set
010305 fluids & plasmas
symbols.namesake
Complex dynamics
Fractal
Modeling and Simulation
Phase space
0103 physical sciences
FOS: Mathematics
symbols
Mathematics - Dynamical Systems
Chaotic Dynamics (nlin.CD)
010306 general physics
Complex plane
Complex quadratic polynomial
Mathematics
Subjects
Details
- ISSN :
- 10075704
- Volume :
- 91
- Database :
- OpenAIRE
- Journal :
- Communications in Nonlinear Science and Numerical Simulation
- Accession number :
- edsair.doi.dedup.....a30bc5722b3f68d136074d72b465c969