Geometric description of chaos in two-degrees-of-freedom Hamiltonian systems

Cerruti-Sola, M.; Pettini, Marco

doi:10.1103/physreve.53.179

Cited by 63 publications

(62 citation statements)

References 10 publications

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“…More recently, we have reconsidered the Riemannian geometric approach and, with the aid of numerical simulations on the FPU-β model, we have discovered why the previous attempts failed: the dominant mechanism for chaotic instability in physically relevant geodesics flows is parametric instability due to curvature variations along the geodesics, and not necessarily geodesic flows on negatively curved manifolds [24,36,37,38,39,40]. On this basis, we have started the formulation of a Riemannian theory of Hamiltonian chaos which applies to dynamical systems described by a standard Lagrangian function…”

Section: Riemannian Geometry Of Chaos In the Fpu-β Modelmentioning

confidence: 99%

“…(13) may exhibit an exponentially growing envelope even if the curvature R(s) is everywhere positive but non constant. For example, in the case of the celebrated Hénon-Heiles model [6], the scalar curvature R, computed with the Jacobi metric, is always positive despite the existence of fully developed chaos above some threshold energy [39]. As a matter of fact, the generic condition of physically relevant systems (like coupled anharmonic oscillators on d-dimensional lattices) is that Ricci and scalar curvatures of the mechanical manifolds are neither constant nor everywhere negative, and the straightforward approach based on Eq.…”

Section: Riemannian Geometry Of Chaos In the Fpu-β Modelmentioning

confidence: 99%

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Weak and strong chaos in Fermi–Pasta–Ulam models and beyond

Pettini

Casetti²,

Cerruti-Sola

et al. 2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We briefly review some of the most relevant results that our group obtained in the past, while investigating the dynamics of the Fermi-Pasta-Ulam (FPU) models. A first result is the numerical evidence of the existence of two different kinds of transitions in the dynamics of the FPU models: i) a Stochasticity Threshold (ST), characterized by a value of the energy per degree of freedom below which the overwhelming majority of the phase space trajectories are regular (vanishing Lyapunov exponents). It tends to vanish as the number N of degrees of freedom is increased. ii) a Strong Stochasticity Threshold (SST), characterized by a value of the energy per degree of freedom at which a crossover appears between two different power laws of the energy dependence of the largest Lyapunov exponent, which phenomenologically corresponds to the transition between weakly and strongly chaotic regimes. It is stable with N . A second result is the development of a Riemannian geometric theory to explain the origin of Hamiltonian chaos. The starting of this theory has been motivated by the inadequacy of the approach based on homoclinic intersections to explain the origin of chaos in systems of arbitrarily large N , or arbitrarily far from quasi-integrability, or displaying a transition between weak and strong chaos. Finally, a third result stems from the search for the transition between weak and strong chaos in systems other than FPU. Actually, we found that a very sharp SST appears as the dynamical counterpart of a thermodynamic phase transition, which in turn has led, in the light of the Riemannian theory of chaos, to the development of a topological theory of phase transitions. "...Fermi expressed often the belief that future fundamental theories in physics may involve non-linear operators and equations, and that it would be useful to attempt practice in the mathematics needed for the understanding of nonlinear systems. The plan was then to start with the possibly simplest such physical model and to study the results of the calculation of its long-time behavior.... The motivation then was to observe the rates of mixing and thermalization with the hope that the computational results would provide hints for a future theory. One could venture a guess that one motive in the selection of problems could be traced to Fermi's early interest in the ergodic theory..."Actually, Fermi's early interest in ergodic theory is witnessed by his contribution to a theorem due to Poincaré and thenceforth known as the Poincaré-Fermi theorem. This asserts that neither analytic (Poincaré) nor smooth (Fermi) integrals of motion besides the energy can survive a generic perturbation of an integrable system with three or more degrees of freedom, thus, in the absence of other isolating integrals of motion, any constant energy surface of these generic systems is expected to be everywhere accessible to the phase space trajectory. At this level, no hindrance to ergodicity seems to be possible. Whence the surprise of Fermi, Pasta and Ulam (FPU) when no apparent...

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Section: Riemannian Geometry Of Chaos In the Fpu-β Modelmentioning

confidence: 99%

Section: Riemannian Geometry Of Chaos In the Fpu-β Modelmentioning

confidence: 99%

Weak and strong chaos in Fermi–Pasta–Ulam models and beyond

Pettini

Casetti²,

Cerruti-Sola

et al. 2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…2.1.3 that it was Krylov who first realized the relevance of mixing for statistical mechanics, and the relevance of the stability properties of geodesics on Riemannian manifolds of negative curvature for mixing. More recently, the geometric approach has been reconsidered with the aid of numerical simulations, finding out that the dominant mechanism for dynamical instability in physically relevant geodesics flows is parametric instability due to curvature variations along the geodesics, instead of the negative curvature [141,142,143,144]. For a dynamical system described by the Lagrangian function…”

Section: Geometric Formalism and The Methods Of Estimating The Largestmentioning

confidence: 99%

“…Now, the microcanonical ensemble averages of K R (q) and of its variance can be computed in terms of the corresponding quantities in the canonical ensemble as follows. The canonical configurational partition function Z(η) is given by 144) where dq = N i=1 dq i . The canonical average K R can of the Ricci curvature K R follows as…”

Section: Geometric Calculation Of λ For Fpu-βmentioning

confidence: 99%

Dynamics of Oscillator Chains

Lichtenberg

Livi

Pettini

et al.

The Fermi-Pasta-Ulam Problem

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Abstract. The Fermi-Pasta-Ulam (FPU) nonlinear oscillator chain has proved to be a seminal system for investigating problems in nonlinear dynamics. First proposed as a nonlinear system to elucidate the foundations of statistical mechanics, the initial lack of confirmation of the researchers expectations eventually led to a number of profound insights into the behavior of high-dimensional nonlinear systems. The initial numerical studies, proposed to demonstrate that energy placed in a single mode of the linearized chain would approach equipartition through nonlinear interactions, surprisingly showed recurrences. Although subsequent work showed that the origin of the recurrences is nonlinear resonance, the question of lack of equipartition remained. The attempt to understand the regularity bore fruit in a profound development in nonlinear dynamics: the birth of soliton theory. A parallel development, related to numerical observations that, at higher energies, equipartition among modes could be approached, was the understanding that the transition with increasing energy is due to resonance overlap. Further numerical investigations showed that time-scales were also important, with a transition between faster and slower evolution. This was explained in terms of mode overlap at higher energy and resonance overlap at lower energy. Numerical limitations to observing a very slow approach to equipartition and the problem of connecting high-dimensional Hamiltonian systems to lower dimensional studies of Arnold diffusion, which indicate transitions from exponentially slow diffusion along resonances to power-law diffusion across resonances, have been considered. Most of the work, both numerical and theoretical, started from low frequency (long wavelength) initial conditions. Coincident with developments to understand equipartition was another program to connect a statistical phenomenon to nonlinear dynamics, that of understanding classical heat conduction. The numerical studies were quite different, involving the excitation of a boundary oscillator with chaotic motion, rather than the excitation of

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“…To this end different metrics are used in literature. Among them, the Eisenhart [16], Finsler [17], Horwitz [18]- [21] or Jacobi [22]- [29] are worth mentioning. We consider the Jacobi metric obtained from the Maupertuis' Principle (1744), where geodesics [30] [31] (natural motions) are identified with trajectories of the system, and the stability can be analyzed through the study of the curvature of this Riemannian manifold via the Jacobi-Levi-Civita equations (JLC) [31].…”

Section: Introductionmentioning

confidence: 99%

Geometrodynamical Analysis to Characterize the Dynamics and Stability of a Molecular System through the Boundary of the Hill’s Region

Vergel

Benito

Losada

et al. 2014

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In this paper we study the dynamics and stability of a two-dimensional model for the vibrations of the LiCN molecule making use of the Riemannian geometry via the Jacobi-Levi-Civita equations applied to the Jacobi metric. The Stability Geometrical Indicator for short times is calculated to locate regular and chaotic trajectories as the relative extrema of this indicator. Only trajectories with initial conditions at the boundary of the Hill's region are considered to characterize the dynamics of the system. The importance of the curvature of this boundary for the stability of trajectories bouncing on it is also discussed.

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Geometric description of chaos in two-degrees-of-freedom Hamiltonian systems

Cited by 63 publications

References 10 publications

Weak and strong chaos in Fermi–Pasta–Ulam models and beyond

Weak and strong chaos in Fermi–Pasta–Ulam models and beyond

Dynamics of Oscillator Chains

Geometrodynamical Analysis to Characterize the Dynamics and Stability of a Molecular System through the Boundary of the Hill’s Region

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