161 research outputs found
On bifurcations of area-preserving and nonorientable maps with quadratic homoclinic tangencies
We study bifurcations of nonorientable area-preserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on nonorientable two-dimensional surfaces. We consider one- and two-parameter general unfoldings and establish results related to the emergence of elliptic periodic orbitPostprint (author’s final draft
Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems
In the present paper we consider the case of a general \cont{r+2}
perturbation, for large enough, of an a priori unstable
Hamiltonian system of degrees of freedom, and we provide
explicit conditions on it, which turn out to be \cont{2} generic
and are verifiable in concrete examples, which guarantee the
existence of Arnold diffusion.
This is a generalization of the result in Delshams et al.,
\emph{Mem. Amer. Math. Soc.}, 2006, where it was considered the case
of a perturbation with a finite number of harmonics in the angular
variables.
The method of proof is based on a careful analysis of the geography
of resonances created by a generic perturbation and it contains a
deep quantitative description of the invariant objects generated by
the resonances therein. The scattering map is used as an essential
tool to construct transition chains of objects of different
topology. The combination of quantitative expressions for both the
geography of resonances and the scattering map provides, in a
natural way, explicit computable conditions for instability
Poincaré-Melnikov-Arnold method for twist maps
The Poincar\'e--Melnikov--Arnold method is the standard tool for detecting splitting of invariant manifolds for systems of ordinary differential equations close to ``integrable'' ones with associated separatrices. This method gives rise to an integral (continuous sum) known as the Melnikov function (or Melnikov integral). If this function is not identically zero, the separatrices split. Moreover, the non-degenerate zeros of this function are associated to transversal intersections of the perturbed invariant (stable and unstable) manifolds. There exists a similar theory for planar maps, and in this case the Melnikov function is not a continuous sum anymore, but an infinite and (a priori) analytically uncomputable (discrete) sum. In a previous work, we have given a method to compute explicitly this kind of sums in terms of elliptic functions, under hypotheses of meromorphicity over the functions in the sum. This method allows us to obtain a strong non-integrability criterion and to apply it to perturbations of elliptic billiards and integrable standard-like maps like the McMillan map. Explicit estimates of the splitting angles are also given. Our aim is extend this method to the study of the splitting of doubly asymptotic manifolds (separatrices) associated to hyperbolic fixed points of twist maps in arbitrary dimensions. We work with maps generated globally by a generating function. Using the variational principle satisfied by these maps, we associate the non-degenerated critical points of a scalar function (here called Melnikov potential) to the transversal intersections of the perturbed asymptotic manifolds. We want to stress the difference of this point of view with the usual one in the literature, that is based in the study of non-degenerated zeros of a vectorial function. The symplectic structure and the variational principle play a fundamental role in our construction. As a first example where this theory can be applied, we study standard-like perturbations of a -dimensional twist map given by~R. McLachlan, for . This map is a multidimensional generalization of the McMillan map. We prove, among other results, that any entire perturbation destroys the separatrix of the McLachlan map
On Birkhoff's conjecture about convex billiards
Birkhoff conjectured that the elliptic billiard was the only integrable convex billiard. Here we prove a local version of this conjecture: any non-trivial symmetric
entire perturbation of an elliptic billiard is non-integrable
Estimates on invariant tori near an elliptic equilibrium point of a Hamiltonian system
We give a precise statement for KAM theorem
in a neighbourhood of an elliptic equilibrium point of a Hamiltonian system.
If the frequencies of the elliptic point are nonresonant up to a certain order
, and a nondegeneracy condition is fulfilled, we get an estimate for
the measure of the complement of the KAM tori in a neighbourhood of given
radius. Moreover, if the frequencies satisfy a Diophantine condition,
with exponent ,
we show that in a neighbourhood of radius the measure of the complement
is exponentially small in .
We also give a related result for quasi-Diophantine frequencies,
which is more useful for practical purposes.
The results are obtained
by putting the system in Birkhoff normal form up to an
appropiate order, and the key point relies on giving accurate bounds for
its terms
Singular separatrix splitting and the Poincare-Melnikov method for area preserving maps
The splitting of separatrices of area preserving maps close to the identity is one of the most paradigmatic examples of an exponentially small or singular phenomenon. The intrinsic small parameter is the characteristic exponent h > 0 of the saddle fixed point. A standard technique to measure the splitting of separatrices is the so-called Poincaré-Melnikov method, which has several specic features in the case of analytic planar maps. The aim of this talk is to compare the predictions for the splitting of separatrices provided by the Poincaré-Melnikov method, with the analytic and numerical results in a simple example where computations in multiple-precision arithmetic are performed
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 manifold.Preprin
Singular separatrix splitting and Melnikov method: An experimental study
We consider families of analytic area-preserving maps depending on two pa-
rameters: the perturbation strength E and the characteristic exponent h of the
origin. For E=0, these maps are integrable with a separatrix to the origin,
whereas they asymptote to flows with homoclinic connections as h->0+. For
fixed E!=0 and small h, we show that these connections break up. The area of
the lobes of the resultant turnstile is given asymptotically by E exp(-Pi^2/h)OÂŞ(h),
where OÂŞ(h) is an even Gevrey-1 function such that OÂŞ(0)!=0 and the radius
of convergence of its Borel transform is
2Pi^2. As E->0 the function OÂŞ tends
to an entire function OÂş. This function OÂş agrees with the one provided by the
Melnikov theory, which cannot be applied directly, due to the exponentially small
size of the lobe area with respect to h.
These results are supported by detailed numerical computations; we use an
expensive multiple-precision arithmetic and expand the local invariant curves up
to very high order
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