Scaling Properties of the Lorenz System and Dissipative Nambu Mechanics

Abstract

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 , acting homogeneously over the whole 3-dim. phase space. In the extended -Lorenz system we find a scaling relation between the dissipation strength 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, ( = 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 → ∞, = 1). In effect weak dissipation, through small values, generates and controls the well explored Route to Chaos in the large r-value regime. The non-dissipative = 0 integrable limit is therefore the gateway to Chaos for the Lorenz system. † The present work is dedicated to the memory of Prof. J.S.Nicolis

Dissipative Nambu systems and oscillator circuit design

“…Note, that the right-hand side corresponds the Jacobian matrix of the vector field f = (f 1 , f 2 , f 2 ) : 3 . The generalized Nambu-Hamiltonian equations of motion or Nambu system with two Nambu-Hamiltonian functions H 1 and H 2 can be formulated as follows…”

“…which represent the non-dissipative part of the Lorenz equations [3,26]. Using the Nambu-Hamiltonian functions…”

“…It should be mentioned that also other authors studied Nambu systems with dissipation; see e.g. [3]. However, in contrast to systems with canonical dissipation the asymptotic dynamic behavior is more complex and and the trajectories cannot be evaluated by in a geometric manner.…”

“…The Lorenz equations with the constant values for σ, r and b allow chaotic behavior (e.g. [31]) that can be understood in more details if the Nambu-Hamiltonian functions and their geometrical interpretation are used; see [3].…”

“…Nambu [23] discussed such a concept that is known as Nambu mechanics, see also [21,32]. More recently, some authors studied the properties to some nonlinear dynamic systems with limit cycles within the framework of the Nambu mechanics [3,26,30]. A preliminary discussion in the context of electrical oscillators is given by [18].…”