84 research outputs found
Analytic invariants associated with a parabolic fixed point in C2
It is well known that in a small neighbourhood of a parabolic fixed point a real-analytic diffeomorphism of (R2,0) embeds in a smooth autonomous flow. In this paper we show that the complex-analytic situation is completely different and a generic diffeomorphism cannot be embedded in an analytic flow in a neighbourhood of its parabolic fixed point. We study two analytic invariants with respect to local analytic changes of coordinates. One of the invariants was introduced earlier by one of the authors. These invariants vanish for time-one maps of analytic flows. We show that one of the invariants does not vanish on an open dense subset. A complete analytic classification of the maps with a parabolic fixed point in C2 is not available at the present time
Oscillating mushrooms: adiabatic theory for a non-ergodic system
Can elliptic islands contribute to sustained energy growth as parameters of a
Hamiltonian system slowly vary with time? In this paper we show that a mushroom
billiard with a periodically oscillating boundary accelerates the particle
inside it exponentially fast. We provide an estimate for the rate of
acceleration. Our numerical experiments confirms the theory. We suggest that a
similar mechanism applies to general systems with mixed phase space.Comment: final revisio
Separatrix splitting at a Hamiltonian bifurcation
We discuss the splitting of a separatrix in a generic unfolding of a
degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We
assume that the unperturbed fixed point has two purely imaginary eigenvalues
and a double zero one. It is well known that an one-parametric unfolding of the
corresponding Hamiltonian can be described by an integrable normal form. The
normal form has a normally elliptic invariant manifold of dimension two. On
this manifold, the truncated normal form has a separatrix loop. This loop
shrinks to a point when the unfolding parameter vanishes. Unlike the normal
form, in the original system the stable and unstable trajectories of the
equilibrium do not coincide in general. The splitting of this loop is
exponentially small compared to the small parameter. This phenomenon implies
non-existence of single-round homoclinic orbits and divergence of series in the
normal form theory. We derive an asymptotic expression for the separatrix
splitting. We also discuss relations with behaviour of analytic continuation of
the system in a complex neighbourhood of the equilibrium
Interpolating vector fields for near indentity maps and averaging
For a smooth near identity map, we introduce the notion of an interpolating vector field written in terms of iterates of the map. Our construction is based on Lagrangian interpolation and provides an explicit expression for autonomous vector fields which approximately interpolate the map. We study properties of the interpolating vector fields and explore their applications to the study of dynamics. In particular, we construct adiabatic invariants for symplectic near identity maps. We also introduce the notion of a Poincaré section for a near identity map and use it to visualise dynamics of four-dimensional maps. We illustrate our theory with several examples, including the Chirikov standard map, a volume-preserving map and a symplectic map in dimension four. The last example is motivated by the theory of Arnold diffusion
On stochastic sea of the standard map
Consider a generic one-parameter unfolding of a homoclinic tangency of an
area preserving surface diffeomorphism. We show that for many parameters
(residual subset in an open set approaching the critical value) the
corresponding diffeomorphism has a transitive invariant set of full
Hausdorff dimension. The set is a topological limit of hyperbolic sets
and is accumulated by elliptic islands.
As an application we prove that stochastic sea of the standard map has full
Hausdorff dimension for sufficiently large topologically generic parameters.Comment: 36 pages, 5 figure
Stable manifolds and homoclinic points near resonances in the restricted three-body problem
The restricted three-body problem describes the motion of a massless particle
under the influence of two primaries of masses and that circle
each other with period equal to . For small , a resonant periodic
motion of the massless particle in the rotating frame can be described by
relatively prime integers and , if its period around the heavier primary
is approximately , and by its approximate eccentricity . We give a
method for the formal development of the stable and unstable manifolds
associated with these resonant motions. We prove the validity of this formal
development and the existence of homoclinic points in the resonant region.
In the study of the Kirkwood gaps in the asteroid belt, the separatrices of
the averaged equations of the restricted three-body problem are commonly used
to derive analytical approximations to the boundaries of the resonances. We use
the unaveraged equations to find values of asteroid eccentricity below which
these approximations will not hold for the Kirkwood gaps with equal to
2/1, 7/3, 5/2, 3/1, and 4/1.
Another application is to the existence of asymmetric librations in the
exterior resonances. We give values of asteroid eccentricity below which
asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3, and 1/2
resonances for any however small. But if the eccentricity exceeds these
thresholds, asymmetric librations will exist for small enough in the
unaveraged restricted three-body problem
Scaling Invariance in a Time-Dependent Elliptical Billiard
We study some dynamical properties of a classical time-dependent elliptical
billiard. We consider periodically moving boundary and collisions between the
particle and the boundary are assumed to be elastic. Our results confirm that
although the static elliptical billiard is an integrable system, after to
introduce time-dependent perturbation on the boundary the unlimited energy
growth is observed. The behaviour of the average velocity is described using
scaling arguments
Shilnikov Lemma for a nondegenerate critical manifold of a Hamiltonian system
We prove an analog of Shilnikov Lemma for a normally hyperbolic symplectic
critical manifold of a Hamiltonian system. Using this
result, trajectories with small energy shadowing chains of homoclinic
orbits to are represented as extremals of a discrete variational problem,
and their existence is proved. This paper is motivated by applications to the
Poincar\'e second species solutions of the 3 body problem with 2 masses small
of order . As , double collisions of small bodies correspond to
a symplectic critical manifold of the regularized Hamiltonian system
Tunneling Mechanism due to Chaos in a Complex Phase Space
We have revealed that the barrier-tunneling process in non-integrable systems
is strongly linked to chaos in complex phase space by investigating a simple
scattering map model. The semiclassical wavefunction reproduces complicated
features of tunneling perfectly and it enables us to solve all the reasons why
those features appear in spite of absence of chaos on the real plane.
Multi-generation structure of manifolds, which is the manifestation of
complex-domain homoclinic entanglement created by complexified classical
dynamics, allows a symbolic coding and it is used as a guiding principle to
extract dominant complex trajectories from all the semiclassical candidates.Comment: 4 pages, RevTeX, 6 figures, to appear in Phys. Rev.
Continuation of the exponentially small transversality for the splitting of separatrices to a whiskered torus with silver ratio
We study the exponentially small splitting of invariant manifolds of
whiskered (hyperbolic) tori with two fast frequencies in nearly-integrable
Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a
torus whose frequency ratio is the silver number . We show
that the Poincar\'e-Melnikov method can be applied to establish the existence
of 4 transverse homoclinic orbits to the whiskered torus, and provide
asymptotic estimates for the tranversality of the splitting whose dependence on
the perturbation parameter satisfies a periodicity property. We
also prove the continuation of the transversality of the homoclinic orbits for
all the sufficiently small values of , generalizing the results
previously known for the golden number.Comment: 17 pages, 2 figure
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