Designable Dynamical Systems for the Generalized Landau Scenario and the Nonlinear Complexification of Periodic Orbits

TL;DR

This paper introduces a method for designing high-dimensional dynamical systems with complex oscillatory behaviors through controlled Hopf bifurcations, enabling rich, scalable, and ordered nonlinear mixing of oscillation modes.

Contribution

It presents a novel system design approach based on linear combinations of variables to achieve nonlinear mixing and complex dynamics without requiring a comprehensive mathematical theory.

Findings

Successfully designed systems exhibit complex oscillatory mixing.

Oscillatory behaviors can be enriched with higher frequency modes while remaining periodic.

The approach is scalable to high-dimensional phase spaces.

Abstract

We have found a way for penetrating the space of the dynamical systems towards systems of arbitrary dimension exhibiting the nonlinear mixing of a large number of oscillation modes through which extraordinarily complex time evolutions arise. The system design is based on assuring the occurrence of a number of Hopf bifurcations in a set of fixed points of a relatively generic system of ordinary differential equations, in which the main peculiarity is that the nonlinearities appear through functions of a linear combination of the system variables. The paper presents the design procedure and a selection of numerical simulations with a variety of designed systems whose dynamical behaviors are really rich and full of unknown features. For concreteness, the presentation is focused to illustrating the oscillatory mixing effects on the periodic orbits, through which the harmonic oscillation born in a Hopf bifurcation becomes successively enriched with the intermittent incorporation of other oscillation modes of higher frequencies while the orbit remains periodic and without necessity of bifurcating instabilities. Even in the absence of a proper mathematical theory covering the nonlinear mixing mechanisms we find enough evidence to expect that the oscillatory scenario be truly scalable concerning the phase space dimension, the multiplicity of involved fixed points and the range of time scales, so that extremely complex but ordered dynamical behaviors could be sustained through it.