Unveiling the fractal structure of Julia sets with Lagrangian descriptors

• Published in: Communications in nonlinear science & numerical simulation Vol. 91; p. 105417
• Main Author: García-Garrido, Víctor J.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.12.2020; Elsevier Science Ltd
• Subjects: Complex dynamics; Diagnostic software; Diagnostic systems; Dynamical systems; Fractals; Julia sets; Lagrange multiplier; Lagrangian descriptors; Newton methods; Phase space structure; Polynomials; Riemann manifold; Lagrangian descriptors; Fractals; Complex dynamics; Julia sets; Phase space structure
• ISSN: 1007-5704; 1878-7274; 1878-7274
• DOI: 10.1016/j.cnsns.2020.105417

•Development of an effective scalar diagnostic to explore the phase space of complex maps.•Analysis of the dynamics generated by archetypal exmaples of rational functions and their Julia sets.•Algorithm capable of revealing simultaneously external rays, equipotentials and laminations. 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 zn+1=zn2+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 tool 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 onto 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.