Nonlinear complexification of periodic orbits in the generalized Landau scenario

We have found a way for penetrating the space of the dynamical systems toward systems of arbitrary dimension exhibiting the nonlinear mixing of a large number of oscillation modes through which extraordinarily complex time evolutions may 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 outlines the design procedure and presents 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 on 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 the 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.

Automated summary

This paper presents a way to penetrate the space of dynamical systems by nonlinearly mixing a large number of oscillation modes, resulting in complex time evolutions. The design is based on the occurrence of Hopf bifurcations in a set of fixed points of ordinary differential equations, where nonlinearities are introduced through functions of a linear combination of system variables. Numerical simulations demonstrate the rich and unknown dynamical behaviors of the designed systems. The paper focuses on the oscillatory mixing effects on periodic orbits, showing the enrichment of harmonic oscillations without bifurcations.