Non Hamiltonian Chaos from Nambu Dynamics of Surfaces

TL;DR

This paper explores how Nambu dynamics of intersecting surfaces can describe non-Hamiltonian chaos, specifically analyzing Lorenz and Rössler attractors through surface foliations and flow decomposition.

Contribution

It introduces a surface-based Nambu dynamics framework to understand dissipative chaos in classical systems like Lorenz and Rössler models.

Findings

Nambu Hamiltonians preserve intersecting surfaces in phase space.

Surface foliations reproduce the dynamics of strange attractors.

Flow decomposition separates volume-preserving and dissipative components.

Abstract

We discuss recent work with E.Floratos (JHEP 1004:036,2010) on Nambu Dynamics of Intersecting Surfaces underlying Dissipative Chaos in $R^{3}$. We present our argument for the well studied Lorenz and R\"{o}ssler strange attractors. We implement a flow decomposition to their equations of motion. Their volume preserving part preserves in time a family of two intersecting surfaces, the so called {\em Nambu Hamiltonians}. For dynamical systems with linear dissipative sector such as the Lorenz system, they are specified in terms of Intersecting Quadratic Surfaces. For the case of the R\"{o}ssler system, with nonlinear dissipative part, they are given in terms of a Helicoid intersected by a Cylinder. In each case they foliate the entire phase space and get deformed by Dissipation, the irrotational component to their flow. It is given by the gradient of a surface in $R^{3}$ specified in terms of a scalar function. All three intersecting surfaces reproduce completely the dynamics of each strange attractor.