42 research outputs found
Interpolating vector fields for near identity 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 expressions 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 Poincar\'e 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 and a symplectic map in dimension four, an example motivated by the theory
of Arnold diffusion.Comment: 28 pages, 9 Figure
Simplified normal forms near a degenerate elliptic fixed point in two-parametric families of area-preserving maps
We derive simplified normal forms for an area-preserving map in a
neighbourhood of a degenerate resonant elliptic fixed point. Such fixed points
appear in generic two-parameter families of area-preserving maps. We also
derive a simplified normal form for a generic two-parametric unfolding. The
normal forms are used to analyse bifurcations of -periodic orbits. In
particular, for we find regions of parameters where the normal form has
"meandering'' invariant curves
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
Arnold diffusion in a priory chaotic symplectic maps
We assume that a symplectic real-analytic map has an invariant normally hyperbolic cylinder and an associated transverse homoclinic cylinder. We prove that generically in the real-analytic category the boundaries of the invariant cylinder are connected by trajectories of the map
Leaky Fermi accelerators
A Fermi accelerator is a billiard with oscillating walls. A leaky accelerator
interacts with an environment of an ideal gas at equilibrium by exchange of
particles through a small hole on its boundary. Such interaction may heat the
gas: we estimate the net energy flow through the hole under the assumption that
the particles inside the billiard do not collide with each other and remain in
the accelerator for sufficiently long time. The heat production is found to
depend strongly on the type of the Fermi accelerator. An ergodic accelerator,
i.e. one which has a single ergodic component, produces a weaker energy flow
than a multi-component accelerator. Specifically, in the ergodic case the
energy gain is independent of the hole size, whereas in the multi-component
case the energy flow may be significantly increased by shrinking the hole size.Comment: 7 pages, 5 figure
Dynamics of 4 symplectic maps near a double resonance
We study the dynamics of a family of symplectic mappings near a doubly resonant elliptic fixed point. We derive and discuss algebraic properties of the resonances required for the analysis of a Takens type normal form. In particular, we propose a classification of the double resonances adapted to this problem, including cases of both strong and weak resonances. Around a weak double resonance (a junction of two resonances of two different orders, both being larger than 4) the dynamics can be described in terms of a simple (in general non-integrable) Hamiltonian model. The non-integrability of the normal form is a consequence of the splitting of the invariant manifolds associated with a normally hyperbolic invariant cylinder. We use a generalisation of the standard map in order to illustrate the difference between a truncated normal form and a full symplectic map. We evaluate numerically the volume of a parallelotope defined by 4 vectors tangent to the stable and unstable manifolds respectively. In good agreement with the general theory this volume is exponentially small with respect to a small parameter and we derive an empirical asymptotic formula which suggests amazing similarity to its analog. Different numerical studies point out that double resonances play a key role to understand Arnold diffusion. This paper has to be seen, also, as a first step in this direction
Equilibration of energy in slow-fast systems
Do partial energies in slowâfast Hamiltonian systems equilibrate? This is a long-standing problem related to the foundation of statistical mechanics. Altering the traditional ergodic assumption, we propose that nonergodicity in the fast subsystem leads to equilibration of the whole system. To show this principle, we introduce a set of mechanical toy modelsâthe springy billiardsâand describe stochastic processes corresponding to their adiabatic behavior. We expect that these models and this principle will play an important role in the quest to establish and study the underlying postulates of statistical mechanics, one of the long-standing scientific grails
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
Splitting of separatrices for (fast) quasiperiodic forcing
We consider fast quasiperiodic perturbations of a pendulum with two frequencies , where is the golden mean number. For small perturbations such that its Fourier coefficients (the ones associated to Fibonacci numbers), are separated from zero, it is announced that the invariant manifolds split, and the value of the splitting, that turns out to be exponentially small with respect to the perturbation parameter, is correctly predicted by the Melnikov function. An explicit example shows that the splitting can be of the order of some power of if the function is not analytic. This makes a qualitative difference between periodic and quasiperiodic perturbation