The applicability of the third integral of motion: Some numerical experiments

Hénon, M.; Heiles, C.

doi:10.1086/109234

Cited by 1,930 publications

(895 citation statements)

References 2 publications

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“…The following aspects are similarly found in the paradigmatic Hénon-Heiles system (Hénon & Heiles 1964): (i) In some regions an adelphic integral of motion is present which hinders the particles on quasiperiodic orbits from escaping.…”

Section: Poincaré Surfaces Of Sectionmentioning

confidence: 83%

Fractal basins of escape and the formation of spiral arms in a galactic potential with a bar

Ernst¹,

Peters²

2014

Monthly Notices of the Royal Astronomical Society

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We investigate the dynamics in the close vicinity of and within the critical area in a 2D effective galactic potential with a bar of Zotos. We have calculated Poincaré surfaces of section and the basins of escape. In both the Poincaré surfaces of section and the basins of escape we find numerical evidence for the existence of a separatrix which hinders orbits from escaping out of the bar region. We present numerical evidence for the similarity between spiral arms of barred spiral galaxies and tidal tails of star clusters.

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Section: Poincaré Surfaces Of Sectionmentioning

confidence: 83%

Fractal basins of escape and the formation of spiral arms in a galactic potential with a bar

Ernst¹,

Peters²

2014

Monthly Notices of the Royal Astronomical Society

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“…2.4.3). The same comparison of the three particle Toda potential with its truncation, the Henon and Heiles potential [11], was very useful in understanding low-dimentional chaos, as we have already discussed briefly in Sect. 2.1.2.…”

Section: Chains Similar To the Fpumentioning

confidence: 92%

“…In particular, the KAM theorem for coupled degrees of freedom [8,9,10] indicated that the generic case was a divided phase space with regular and chaotic orbits interspersed. Numerical observations, in a surface of section of a particular two degree of freedom system (the Hénon and Heiles potential), indicated mostly regular orbits at low energy, with the chaotic portion of the phase space increasing rather abruptly over a small range of increasing energy, until most of the phase space is chaotic [11]. A practical explanation of this rather abrupt increase was that local resonances between frequencies of the two freedoms, which modified the structure of the phase space in their neighborhood, would overlap with increasing energy, producing large areas of chaotic motion [12,13].…”

Section: Chaos Theory: Kam Isolation Arnold Diffusion Lyapunov Expomentioning

confidence: 98%

“…In addition to the energy, there is a relatively obvious isolating integral, namely the total momentum, reducing the motion to two degrees of freedom. The Henon-Heiles potential [11] is a truncation of the Toda lattice. However, a surface of section of (2.4), calculated numerically, showed no chaos [21] and Hamiltonian (2.4) was subsequently proved to have a third invariant and thus was integrable, i.e.…”

Section: Chaos Theory: Kam Isolation Arnold Diffusion Lyapunov Expomentioning

confidence: 99%

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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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“…Let us just give one example [31]. The Hamiltonian Hénon-Heiles system [62] in two coupled variables (q 1 , q 2 ) The diophantine equations to be solved are Three of them restrict the ODE to the stationary reduction of well-known soliton equations, thus proving the PP : Sawada-Kotera (SK [103]), higherorder Korteweg-de Vries (KdV5, [80]) and Kaup-Kupershmidt (KK, [71,51]) equations.…”

Section: The Diophantine Conditionsmentioning

confidence: 99%

The Painlevé Approach to Nonlinear Ordinary Differential Equations

Conte¹

1999

The Painlevé Property

166

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The "Painlevé analysis" is quite often perceived as a collection of tricks reserved to experts. The aim of this course is to demonstrate the contrary and to unveil the simplicity and the beauty of a subject which is in fact the theory of the (explicit) integration of nonlinear differential equations.To achieve our goal, we will not start the exposition with a more or less precise "Painlevé test". On the contrary, we will finish with it, after a gradual introduction to the rich world of singularities of nonlinear differential equations, so as to remove any cooking recipe.The emphasis is put on embedding each method of the test into the well known theorem of perturbations of Poincaré. A summary can be found at the beginning of each chapter.

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The applicability of the third integral of motion: Some numerical experiments

Abstract: The problem of the existence of a third isolating integral of motion in an axisymmetric potential is investigated by numerical experiments. It is found that the third integral exists for only a limited range of initial conditions.

Cited by 1,930 publications

References 2 publications

Fractal basins of escape and the formation of spiral arms in a galactic potential with a bar

Fractal basins of escape and the formation of spiral arms in a galactic potential with a bar

Dynamics of Oscillator Chains

The Painlevé Approach to Nonlinear Ordinary Differential Equations

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