Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2011-09-02
Chaos Theory: Modeling, Simulation and Applications, C.H. Skiadas, I. Dimotikalis and C. Skiadas (Eds), World Scientific Publi
Nonlinear Sciences
Chaotic Dynamics
10 pages, Invited Talks at the International Conferences on: Nonlinear Dynamics and Complexity; Theory, Methods and Applicatio
Scientific paper
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.
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