Phenomena of complex analytic dynamics in the systems of alternately excited coupled non-autonomous oscillators and self-sustained oscillators

TL;DR

This paper introduces a model of coupled oscillators that exhibits phenomena similar to complex analytic maps, such as Mandelbrot and Julia sets, and analyzes how non-analytic terms affect these structures.

Contribution

The study presents a novel oscillator system that replicates complex analytic dynamics and investigates the impact of non-analytic terms on fractal structures.

Findings

Mandelbrot-like set rotation observed in parameter space

Destruction of small-scale fractal structures due to non-analytic terms

Analytic and numerical analysis of complex dynamics in coupled oscillators

Abstract

A feasible model is introduced that manifests phenomena intrinsic to iterative complex analytic maps (such as the Mandelbrot set and Julia sets). The system is composed of two coupled alternately excited oscillators (or self-sustained oscillators). The idea is based on a turn-by-turn transfer of the excitation from one subsystem to another (S.P.~Kuznetsov, Phys.~Rev.~Lett. \bf 95 \rm, 2005, 144101) accompanied with appropriate nonlinear transformation of the complex amplitude of the oscillations in the course of the process. Analytic and numerical studies are performed. Special attention is paid to an analysis of the violation of the applicability of the slow amplitude method with the decrease in the ratio of the period of the excitation transfer to the basic period of the oscillations. The main effect is the rotation of the Mandelbrot-like set in the complex parameter plane; one more effect is the destruction of subtle small-scale fractal structure of the set due to the presence of non-analytic terms in the complex amplitude equations.