Emergence of multi-scroll attractors

Abstract Phase space trajectories are fundamentally important for understanding and analysing chaotic attractors. Tools for characterising these chaotic attractors take their input from the direct numerical solution of the dynamical equations. Properties of scrolls can also be characterised by these tools. Though the topological properties of scrolls can be understood using tools like topological template and the linking number, their emergence in the phase space can also directly be inferred from the geometry and relative orientations of Nambu surfaces (drawn using Nambu Hamiltonians) rather than from direct numerical solutions. These energy-like Hamiltonians can be extracted from the vector flow of the governing equations. Therefore, one can attribute the origin of scrolls in phase space to energy-like Nambu surfaces, providing a geometrical interpretation. In this article, we have carried out both numerical analysis of the bifurcation diagram and Lyapunov exponents to characterise chaos, and then applied the Nambu generalised Hamiltonian mechanics to explain the fundamental reason for the appearance of scroll-like geometry in the phase space. We have demonstrated how a fixed number of scrolls can appear in the phase space due to the specific geometry of the Nambu surfaces.