Analysis of Butterfly Effect using Nambu Formalism

Abstract

Nonlinear dynamics is at the heart of the modern interdisciplinary approach to science. In this article, chaotic Fourwing attractor is investigated by means of some of the diagnostic tools such as bifurcation diagram, Lyapunov characteristic exponents, Poincar´e maps to measure the extent of chaos . Here, a 3-dimensional autonomous chaotic Fourwing attractor is analyzed with the help of phase-space geometry using Nambu mechanics. From geometrical point of view, it is observed that with the help of Nambu mechanics the non-dissipative part of the chaotic system can be generated by the intersection of two quadratic surfaces that form a Nambu doublet. All manifolds are classified into four distinct classes; parabolic, hyperbolic, cylindrical and elliptical. By using different linear transformations with Jacobian J=1, we can find different classes of doublets associated with the chaotic system. The boundaries of the attractor is also found. Another interesting fact we find that this Nambu mechanics framework can be extended to include dissipation in R3 phase-space. We demonstrate that the dissipative Nambu dynamics give rise to the intersecting surfaces of the strange attractor in a very intuitive manner accounting their gross topological aspects.