Emergence of Multi-Scroll Attractors

TL;DR

This paper explores the geometric origins of multi-scroll attractors in chaotic systems by analyzing phase space trajectories, bifurcations, and Nambu surfaces, offering a new perspective beyond direct numerical solutions.

Contribution

It introduces a geometric approach using Nambu Hamiltonians to explain the emergence of multi-scroll attractors, linking phase space wings to Nambu surface geometry.

Findings

Multi-scroll attractors are linked to specific Nambu surface geometries.

Parameter changes alter the number and shape of scrolls.

Lyapunov exponents confirm chaotic behavior.

Abstract

Phase space trajectories are fundamentally important for understanding and analysing chaotic attractors. This is mostly carried out by direct numerical solution of the dynamical equations. Though the origin of scrolls can be understood from the properties of dynamical equations, their appearance in the phase space can also be inferred from the geometry and relative orientations of Nambu surfaces, drawn using Nambu Hamiltonians than from direct numerical solutions. Therefore, one can attribute the origin of wings in the phase space due to energy like Nambu surfaces, giving a geometrical interpretation. In this article, we have carried out, both numerical analysis of bifurcation diagram and Lyapunov exponents(LEs) to characterise chaos and geometric approach by applying the Nambu generalized Hamiltonian mechanics to explain the fundamental reason for the appearance of wings like geometry in the phase space. We have shown how a fixed number of scrolls or wings can appear in the phase space due to specific geometry of the Nambu surfaces and how different geometries are formed when set of parameters are changed.