59 research outputs found
Lyapunov exponents from geodesic spread in configuration space
The exact form of the Jacobi–Levi-Civita (JLC) equation for geodesic spread is here explicitly worked out at arbitrary dimension for the configuration space manifold [Formula Presented] of a standard Hamiltonian system, equipped with the Jacobi (or kinetic energy) metric [Formula Presented] As the Hamiltonian flow corresponds to a geodesic flow on [Formula Presented] the JLC equation can be used to study the degree of instability of the Hamiltonian flow. It is found that the solutions of the JLC equation are closely resembling the solutions of the standard tangent dynamics equation which is used to compute Lyapunov exponents. Therefore the instability exponents obtained through the JLC equation are in perfect quantitative agreement with usual Lyapunov exponents. This work completes a previous investigation that was limited only to two degrees of freedom systems. © 1997 The American Physical Society
Lyapunov exponents from geodesic spread in configuration space
The exact form of the Jacobi -- Levi-Civita (JLC) equation for geodesic
spread is here explicitly worked out at arbitrary dimension for the
configuration space manifold M_E = {q in R^N | V(q) < E} of a standard
Hamiltonian system, equipped with the Jacobi (or kinetic energy) metric g_J. As
the Hamiltonian flow corresponds to a geodesic flow on (M_E,g_J), the JLC
equation can be used to study the degree of instability of the Hamiltonian
flow. It is found that the solutions of the JLC equation are closely resembling
the solutions of the standard tangent dynamics equation which is used to
compute Lyapunov exponents. Therefore the instability exponents obtained
through the JLC equation are in perfect quantitative agreement with usual
Lyapunov exponents. This work completes a previous investigation that was
limited only to two-degrees of freedom systems.Comment: REVTEX file, 10 pages, 2 figure
Weak and strong chaos in Fermi-Pasta-Ulam models and beyond
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. The 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 weak and strong chaotic regimes. It is stable with N. The second result is the development of a Riemannian geometric theory to explain the origin of Hamiltonian chaos. Starting 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, the 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. (C) 2005 American Institute of Physics
The gas turbulence in planetary nebulae: quantification and multi-D maps from long-slit, wide-spectral range echellogram
This methodological paper is part of a short series dedicated to the
long-standing astronomical problem of de-projecting the bi-dimensional,
apparent morphology of a three-dimensional distribution of gas. We focus on the
quantification and spatial recovery of turbulent motions in planetary nebulae
(and other classes of expanding nebulae) by means of long-slit echellograms
over a wide spectral range. We introduce some basic theoretical notions,
discuss the observational methodology, and develop an accurate procedure
disentangling all broadening components of the velocity profile in all spatial
positions of each spectral image. This allows us to extract random, non-thermal
motions at unprecedented accuracy, and to map them in 1-, 2- and 3-dimensions.
We present the solution to practical problems in the multi-dimensional
turbulence-analysis of a testing-planetary nebula (NGC 7009), using the
three-step procedure (spatio-kinematics, tomography, and 3-D rendering)
developed at the Astronomical Observatory of Padua. In addition, we introduce
an observational paradigm valid for all spectroscopic parameters in all classes
of expanding nebulae. Unsteady, chaotic motions at a local scale constitute a
fundamental (although elusive) kinematical parameter of each planetary nebula,
providing deep insights on its different shaping agents and mechanisms, and on
their mutual interaction. The detailed study of turbulence, its stratification
within a target and (possible) systematic variation among different sub-classes
of planetary nebulae deserve long-slit, multi-position angle, wide-spectral
range echellograms containing emissions at low-, medium-, and high-ionization,
to be analyzed pixel-to-pixel with a straightforward and versatile methodology,
extracting all the physical information stored in each frame at best.Comment: 11 page, 10 figures, A&A in pres
UV (IUE) spectra of the central stars of high latitude planetary nebulae Hb7 and Sp3
We present an analysis of the UV (IUE) spectra of the central stars of Hb7
and Sp3. Comparison with the IUE spectrum of the standard star HD 93205 leads
to a spectral classification of O3V for these stars, with an effective
temperature of 50,000 K. From the P-Cygni profiles of CIV (1550 A), we derive
stellar wind velocities and mass loss rates of -1317 km/s +/- 300 km/s and
2.9X10^{-8} solar mass yr^{-1} and -1603 km/s +/- 400 km/s and 7X10^{-9} solar
mass yr^{-1} for Hb7 and Sp3 respectively. From all the available data, we
reconstruct the spectral energy distribution of Hb7 and Sp3.Comment: 4 pages, 3 figures, latex, accepted for publication in Astronomy &
Astrophysic
Hamiltonian dynamics and geometry of phase transitions in classical XY models
The Hamiltonian dynamics associated to classical, planar, Heisenberg XY
models is investigated for two- and three-dimensional lattices. Besides the
conventional signatures of phase transitions, here obtained through time
averages of thermodynamical observables in place of ensemble averages,
qualitatively new information is derived from the temperature dependence of
Lyapunov exponents. A Riemannian geometrization of newtonian dynamics suggests
to consider other observables of geometric meaning tightly related with the
largest Lyapunov exponent. The numerical computation of these observables -
unusual in the study of phase transitions - sheds a new light on the
microscopic dynamical counterpart of thermodynamics also pointing to the
existence of some major change in the geometry of the mechanical manifolds at
the thermodynamical transition. Through the microcanonical definition of the
entropy, a relationship between thermodynamics and the extrinsic geometry of
the constant energy surfaces of phase space can be naturally
established. In this framework, an approximate formula is worked out,
determining a highly non-trivial relationship between temperature and topology
of the . Whence it can be understood that the appearance of a phase
transition must be tightly related to a suitable major topology change of the
. This contributes to the understanding of the origin of phase
transitions in the microcanonical ensemble.Comment: in press on Physical Review E, 43 pages, LaTeX (uses revtex), 22
PostScript figure
Absolute properties of the binary system BB Pegasi
We present a ground based photometry of the low-temperature contact binary BB
Peg. We collected all times of mid-eclipses available in literature and
combined them with those obtained in this study. Analyses of the data indicate
a period increase of 3.0(1) x 10^{-8} days/yr. This period increase of BB Peg
can be interpreted in terms of the mass transfer 2.4 x 10^{-8} Ms yr^{-1} from
the less massive to the more massive component. The physical parameters have
been determined as Mc = 1.42 Ms, Mh = 0.53 Ms, Rc = 1.29 Rs, Rh = 0.83 Rs, Lc =
1.86 Ls, and Lh = 0.94 Ls through simultaneous solution of light and of the
radial velocity curves. The orbital parameters of the third body, that orbits
the contact system in an eccentric orbit, were obtained from the period
variation analysis. The system is compared to the similar binaries in the
Hertzsprung-Russell and Mass-Radius diagram.Comment: 17 pages, 3 figures, accepted for Astronomical Journa
Riemannian theory of Hamiltonian chaos and Lyapunov exponents
This paper deals with the problem of analytically computing the largest
Lyapunov exponent for many degrees of freedom Hamiltonian systems. This aim is
succesfully reached within a theoretical framework that makes use of a
geometrization of newtonian dynamics in the language of Riemannian geometry. A
new point of view about the origin of chaos in these systems is obtained
independently of homoclinic intersections. Chaos is here related to curvature
fluctuations of the manifolds whose geodesics are natural motions and is
described by means of Jacobi equation for geodesic spread. Under general
conditions ane effective stability equation is derived; an analytic formula for
the growth-rate of its solutions is worked out and applied to the
Fermi-Pasta-Ulam beta model and to a chain of coupled rotators. An excellent
agreement is found the theoretical prediction and the values of the Lyapunov
exponent obtained by numerical simulations for both models.Comment: RevTex, 40 pages, 8 PostScript figures, to be published in Phys. Rev.
E (scheduled for November 1996
Collapses and explosions in self-gravitating systems
Collapse and reverse to collapse explosion transition in self-gravitating
systems are studied by molecular dynamics simulations. A microcanonical
ensemble of point particles confined to a spherical box is considered; the
particles interact via an attractive soft Coulomb potential. It is observed
that the collapse in the particle system indeed takes place when the energy of
the uniform state is put near or below the metastability-instability threshold
(collapse energy), predicted by the mean-field theory. Similarly, the explosion
in the particle system occurs when the energy of the core-halo state is
increased above the explosion energy, where according to the mean field
predictions the core-halo state becomes unstable. For a system consisting of
125 -- 500 particles, the collapse takes about single particle crossing
times to complete, while a typical explosion is by an order of magnitude
faster. A finite lifetime of metastable states is observed. It is also found
that the mean-field description of the uniform and the core-halo states is
exact within the statistical uncertainty of the molecular dynamics data.Comment: 9 pages, 14 figure
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