Strange attractors in dissipative Nambu mechanics: classical and quantum aspects

Axenides, Minos; Floratos, E.G.

doi:10.1007/jhep04(2010)036

Cited by 10 publications

(19 citation statements)

References 75 publications

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“…By fixing r in the -Lorenz system and taking the limit → 0 we recover the nondissipative Lorenz system of eq.(1.4). It is integrable and describes the motion of a particle in an one dimensional anharmonic potential [1]( or the pendulum). We can still rescale this sytem by choosing λ = 1 √ r and we discover the infinite r limit of the full Lorenz system, which has been studied in detail in ref.…”

Section: Some Remarks Are In Odermentioning

confidence: 99%

“…We choose to work with the Cylinder intersecting with a Paraboloid as the two Nambu "Hamiltonians" among the whole set of SL(2,R) geometries [1].…”

Section: Figure 2: Lorenz ζ Dissipation Parametermentioning

confidence: 99%

“…In the framework of Nambu mechanics we have recently [1,2] proposed a geometric approach for the study of dynamical dissipative systems [3]. It is implemented through a special decomposition of the flow of the system in two parts.…”

Section: Introductionmentioning

confidence: 99%

“…The non-dissipative Lorenz system which was studied carefully in [1] was also investigated some time ago in [7] and in a similar vein further back in time by [8] ˙ x = ∇H 1 × ∇H 2 (1.4) is an integrable dynamical system, where the trajectories in phase space are given by the intersection of the surfaces defined by the conserved Nambu " Hamiltonians" (H 1 , H 2 ) . In this picture the dynamical system in rel(1.1) is defined through its non-dissipative part of eq,(1.4), over which its phase space dissipation ∇D operates.…”

Section: Introductionmentioning

confidence: 99%

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Scaling Properties of the Lorenz System and Dissipative Nambu Mechanics

Axenides

Floratos

2014

Chaos, Information Processing and Paradoxical Games

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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

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Section: Some Remarks Are In Odermentioning

confidence: 99%

“…We choose to work with the Cylinder intersecting with a Paraboloid as the two Nambu "Hamiltonians" among the whole set of SL(2,R) geometries [1].…”

Section: Figure 2: Lorenz ζ Dissipation Parametermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Scaling Properties of the Lorenz System and Dissipative Nambu Mechanics

Axenides

Floratos

2014

Chaos, Information Processing and Paradoxical Games

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“…Névir and Blender [22] developed a Nambu representation of the nondissipative parts of the classical Lorenz-63 [12] model for X 1 , Y 2 , and Z 1 (see also [1,24,3] and [25] for a classification). With the conservation laws for H and C the Lorenz equations are in terms of x = (X 1 , Y 2 , Z 1 )…”

Section: Nambu Structurementioning

confidence: 99%

Nambu representation of an extended Lorenz model with viscous heating

Blender¹,

Lucarini²

2013

Physica D: Nonlinear Phenomena

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We consider the Nambu and Hamiltonian representations of RayleighBénard convection with a nonlinear thermal heating effect proportional to the Eckert number (Ec). The model we use is an extension of the classical Lorenz-63 model with 4 kinematic and 6 thermal degrees of freedom. The conservative parts of the dynamical equations which include all nonlinearities satisfy Liouville's theorem and permit a conserved Hamiltonian H for arbitrary Ec. For Ec = 0 two independent conserved Casimir functions exist, one of these is associated with unavailable potential energy and is also present in the Lorenz-63 truncation. This Casimir C is used to construct a Nambu representation of the conserved part of the dynamical system. The thermal heating effect can be represented either by a second canonical Hamiltonian or as a gradient (metric) system using the time derivativeĊ of the Casimir. The results demonstrate the impact of viscous heating in the total energy budget and in the Lorenz energy cycle for kinetic and available potential energy.

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Nambu mechanics for stochastic magnetization dynamics

Thibaudeau

Nussle

2017

Journal of Magnetism and Magnetic Materials

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The Landau-Lifshitz-Gilbert (LLG) equation describes the dynamics of a damped magnetization vector that can be understood as a generalization of Larmor spin precession. The LLG equation cannot be deduced from the Hamiltonian framework, by introducing a coupling to a usual bath, but requires the introduction of additional constraints. It is shown that these constraints can be formulated elegantly and consistently in the framework of dissipative Nambu mechanics. This has many consequences for both the variational principle and for topological aspects of hidden symmetries that control conserved quantities. We particularly study how the damping terms of dissipative Nambu mechanics affect the consistent interaction of magnetic systems with stochastic reservoirs and derive a master equation for the magnetization. The proposals are supported by numerical studies using symplectic integrators that preserve the topological structure of Nambu equations. These results are compared to computations performed by direct sampling of the stochastic equations and by using closure assumptions for the moment equations, deduced from the master equation.

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Strange attractors in dissipative Nambu mechanics: classical and quantum aspects

Cited by 10 publications

References 75 publications

Scaling Properties of the Lorenz System and Dissipative Nambu Mechanics

Scaling Properties of the Lorenz System and Dissipative Nambu Mechanics

Nambu representation of an extended Lorenz model with viscous heating

Nambu mechanics for stochastic magnetization dynamics

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