Scaling Properties of the Lorenz System and Dissipative Nambu Mechanics

In the framework of Nambu Mechanics, we have recently argued that Non-Hamiltonian Chaotic Flows in $ R^{3} $, are dissipation induced deformations, of integrable volume preserving flows, specified by pairs of Intersecting Surfaces in $R^{3}$. In the present work we focus our attention to the Lorenz system with a linear dissipative sector in its phase space dynamics. In this case the Intersecting Surfaces are Quadratic. We parametrize its dissipation strength through a continuous control parameter $\epsilon$, acting homogeneously over the whole 3-dim. phase space. In the extended $\epsilon$-Lorenz system we find a scaling relation between the dissipation strength $ \epsilon $ and Reynolds number parameter r . It results from the scale covariance, we impose on the Lorenz equations under arbitrary rescalings of all its dynamical coordinates. Its integrable limit, ($ \epsilon = 0 $, \ fixed r), which is described in terms of intersecting Quadratic Nambu "Hamiltonians" Surfaces, gets mapped on the infinite value limit of the Reynolds number parameter (r $\rightarrow \infty,\ \epsilon= 1$). In effect weak dissipation, through small $\epsilon$ values, generates and controls the well explored Route to Chaos in the large r-value regime. The non-dissipative $\epsilon=0 $ integrable limit is therefore the gateway to Chaos for the Lorenz system.