Emergence of Multi-Scroll Attractors

TL;DR

This paper addresses how multiple scrolls or wings arise in a 3D chaotic attractor, proposing a geometric interpretation based on intersecting Nambu Hamiltonian surfaces rather than solely on numerical integration. The authors apply Nambu mechanics with a two-Hamiltonian formulation, decomposing the dynamics into a volume-preserving non-dissipative part $\mathbf{v}_{ND}=\nabla H_1 \times \nabla H_2$ and a dissipative part $\mathbf{v}_D=\nabla D$, and derive explicit $H_1$, $H_2$, and $D$ for a Wang-type multi-scroll system. Intersections of the Nambu surfaces $H_1=k_1$, $H_2=k_2$ define non-dissipative scroll orbits, while the inclusion of dissipation yields full chaotic attractors with the number of scrolls controlled by system parameters $a$ and $d$. Bifurcation analysis and Lyapunov exponents map chaotic regimes, linking geometry to dynamics and offering a framework for predicting and potentially steering multi-scroll chaos.

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.

Figures (11)

• Figure 1: Phase-space trajectories of different multi-scroll attractors emerging from Eq.(\ref{['eq1']}) for various set of parameter values and initial conditions. In each sub-figure, (a-d)(i) represents 3D phase-space trajectories and (a-d)(ii), (a-d)(iii), (a-d)(iv) represents 2D phase-space trajectories in x-y plane, y-z plane, z-x plane respectively.
• Figure 2: It represents the bifurcation diagram, indicating the system's dynamics along x-axis for control parameter $“b”$ varying within the range $0 < b < 12$ by keeping $a=1, c=5, d=0.06$ fixed. And the inset figures illustrates the existence of multi-scroll attractors in the system(\ref{['eq1']}) by varying the control parameter $"b"$ such as a point attractor for $b=1$, 1-scroll limit cycle for $b=1.5$, period-2 orbit for $b=1.58$, period-4 orbit for $b=1.63$, 1-scroll chaotic attractor for $b=1.67$, 2-scroll chaotic attractor for $b=3.2$, 3-scroll chaotic attractor for $b=9$, 4-scroll chaotic attractor for $b=9.8$.
• Figure 3: It illustrates that the multi-scroll attractors have multiple equilibrium points and each "scroll" corresponds to a region around an equilibrium point. Here, Red colour points indicate the equilibrium points of corresponding n-scroll attractor.
• Figure 4: Bifurcation diagrams in (a) and (c) indicate the existence of corresponding chaotic regimes for the individual variation of control parameter "a" and "d" respectively. The corresponding Lyapunov exponent spectrums are shown in (b) and (d).
• Figure 5: Density plot of maximum Lyapunov exponents in a-d parameter space.