On possibility of realization of the Mandelbrot set in coupled continuous systems

TL;DR

This paper explores the potential to realize Mandelbrot and Julia sets in coupled continuous systems, highlighting challenges and analyzing a specific example of coupled R"{o}ssler oscillators.

Contribution

It investigates the feasibility of generating complex analytic dynamics in coupled autonomous continuous systems, extending previous methods to new system types.

Findings

Difficulties in realizing Mandelbrot sets in coupled autonomous continuous systems

Analysis of coupled R"{o}ssler oscillators as an example

Discussion on differences from discrete maps and driven systems

Abstract

According to the method, suggested in our previous work (nlin/0509012) and based on the consideration of the specially coupled systems, the possibility of physical realization of the phenomena of complex analytic dynamics (such as Mandelbrot and Julia sets) is discussed. It is shown, that unlike the case of discrete maps or differential systems with periodic driving, investigated in mentioned work, there are some difficulties in attempts to obtain the Mandelbrot set for the coupled autonomous continuous systems. A system of coupled autonomous R\"{o}ssler oscillators is considered as an example.