1,933 research outputs found
Parabolic resonances and instabilities in near-integrable two degrees of freedom Hamiltonian flows
When an integrable two-degrees-of-freedom Hamiltonian system possessing a
circle of parabolic fixed points is perturbed, a parabolic resonance occurs. It
is proved that its occurrence is generic for one parameter families
(co-dimension one phenomenon) of near-integrable, t.d.o. systems. Numerical
experiments indicate that the motion near a parabolic resonance exhibits new
type of chaotic behavior which includes instabilities in some directions and
long trapping times in others. Moreover, in a degenerate case, near a {\it flat
parabolic resonance}, large scale instabilities appear. A model arising from an
atmospherical study is shown to exhibit flat parabolic resonance. This supplies
a simple mechanism for the transport of particles with {\it small} (i.e.
atmospherically relevant) initial velocities from the vicinity of the equator
to high latitudes. A modification of the model which allows the development of
atmospherical jets unfolds the degeneracy, yet traces of the flat instabilities
are clearly observed
Aspects of the planetary Birkhoff normal form
The discovery in [G. Pinzari. PhD thesis. Univ. Roma Tre. 2009], [L.
Chierchia and G. Pinzari, Invent. Math. 2011] of the Birkhoff normal form for
the planetary many--body problem opened new insights and hopes for the
comprehension of the dynamics of this problem. Remarkably, it allowed to give a
{\sl direct} proof of the celebrated Arnold's Theorem [V. I. Arnold. Uspehi
Math. Nauk. 1963] on the stability of planetary motions. In this paper, using a
"ad hoc" set of symplectic variables, we develop an asymptotic formula for this
normal form that may turn to be useful in applications. As an example, we
provide two very simple applications to the three-body problem: we prove a
conjecture by [V. I. Arnold. cit] on the "Kolmogorov set"of this problem and,
using Nehoro{\v{s}}ev Theory [Nehoro{\v{s}}ev. Uspehi Math. Nauk. 1977], we
prove, in the planar case, stability of all planetary actions over
exponentially-long times, provided mean--motion resonances are excluded. We
also briefly discuss perspectives and problems for full generalization of the
results in the paper.Comment: 44 pages. Keywords: Averaging Theory, Birkhoff normal form,
Nehoro{\v{s}}ev Theory, Planetary many--body problem, Arnold's Theorem on the
stability of planetary motions, Properly--degenerate kam Theory, steepness.
Revised version, including Reviewer's comments. Typos correcte
A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems
In this paper we consider a representative a priori unstable Hamiltonian
system with 2+1/2 degrees of freedom, to which we apply the geometric mechanism
for diffusion introduced in the paper Delshams et al., Mem. Amer. Math. Soc.
2006, and generalized in Delshams and Huguet, Nonlinearity 2009, and provide
explicit, concrete and easily verifiable conditions for the existence of
diffusing orbits.
The simplification of the hypotheses allows us to perform explicitly the
computations along the proof, which contribute to present in an easily
understandable way the geometric mechanism of diffusion. In particular, we
fully describe the construction of the scattering map and the combination of
two types of dynamics on a normally hyperbolic invariant manifol
Foliations of Isonergy Surfaces and Singularities of Curves
It is well known that changes in the Liouville foliations of the isoenergy
surfaces of an integrable system imply that the bifurcation set has
singularities at the corresponding energy level. We formulate certain
genericity assumptions for two degrees of freedom integrable systems and we
prove the opposite statement: the essential critical points of the bifurcation
set appear only if the Liouville foliations of the isoenergy surfaces change at
the corresponding energy levels. Along the proof, we give full classification
of the structure of the isoenergy surfaces near the critical set under our
genericity assumptions and we give their complete list using Fomenko graphs.
This may be viewed as a step towards completing the Smale program for relating
the energy surfaces foliation structure to singularities of the momentum
mappings for non-degenerate integrable two degrees of freedom systems.Comment: 30 pages, 19 figure
- …