Phase space geometry of a Four-wings chaotic attractor

TL;DR

This paper investigates the phase space geometry of a four-wings chaotic attractor, revealing how complex structures arise and characterizing the conditions for their formation using numerical and analytical methods.

Contribution

It introduces a detailed analysis of four-wings chaotic attractors, combining numerical solutions and Nambu mechanics to understand their geometric and dynamical properties.

Findings

Four-wings structure arises in phase space of chaotic systems.

Attractors form from intersections of Hamiltonian Nambu functions.

Analytical conditions for localized attractor regions are derived.

Abstract

The well known butterfly effect got its nomenclature from its two wings geometrical structure in phase space. There are chaotic dynamics from simple one-wing to multiple-wings complex structures in phase space. In this communication we demonstrate, both with direct numerical solutions and using Nambu mechanics, how does a four-wings complex structure in the phase space arise for a chaotic dynamical system. We further explore the properties of these structures and demonstrate that an attractor is produced out of dynamical intersections of Hamiltonian kind of Nambu functions. We also find, analytically, the specific conditions on system parameters for the formation of localized region of an attractor.