Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies

We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional torus with a fast frequency vector ¿/ev, with ¿=(1,O,O˜) where O is a cubic irrational number whose two conjugates are complex, and the components of ¿ generate the field Q(O). A paradigmatic case is the cubic golden vector, given by the (real) number O satisfying O3=1-O, and O˜=O2. For such 3-dimensional frequency vectors, the standard theory of continued fractions cannot be applied, so we develop a methodology for determining the behavior of the small divisors ¿k,¿¿, k¿Z3. Applying the Poincaré–Melnikov method, this allows us to carry out a careful study of the dominant harmonic (which depends on e) of the Melnikov function, obtaining an asymptotic estimate for the maximal splitting distance, which is exponentially small in e, and valid for all sufficiently small values of e. This estimate behaves like exp{-h1(e)/e1/6} and we provide, for the first time in a system with 3 frequencies, an accurate description of the (positive) function h1(e) in the numerator of the exponent, showing that it can be explicitly constructed from the resonance properties of the frequency vector ¿, and proving that it is a quasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence of the estimates for the splitting on the arithmetic properties of the frequencies.Peer ReviewedPreprin

Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows

AbstractWe show that certain mechanical systems, including a geodesic flow in any dimension plus a quasi-periodic perturbation by a potential, have orbits of unbounded energy.The assumptions we make in the case of geodesic flows are:(a)The metric and the external perturbation are smooth enough.(b)The geodesic flow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection.(c)The frequency of the external perturbation is Diophantine.(d)The external potential satisfies a generic condition depending on the periodic orbit considered in (b).The assumptions on the metric are C2 open and are known to be dense on many manifolds. The assumptions on the potential fail only in infinite codimension spaces of potentials.The proof is based on geometric considerations of invariant manifolds and their intersections. The main tools include the scattering map of normally hyperbolic invariant manifolds, as well as standard perturbation theories (averaging, KAM and Melnikov techniques).We do not need to assume that the metric is Riemannian and we obtain results for Finsler or Lorentz metrics. Indeed, there is a formulation for Hamiltonian systems satisfying scaling hypotheses. We do not need to make assumptions on the global topology of the manifold nor on its dimension

Instabilities in Hamiltonian systems

In 1964, V. I. Arnol'd proved the existence of nearly-integrable Hamiltonian systems which have global instabilities (global chaotic behaviour). This phenomenon is nowadays termed under the name "Arnol'd diffusion". One of the key ideas that he used is to "travel" along invariant manifolds of the Hamiltonian system. The purpose of this project is to understand the Arnol'd instability mechanism and study new ones using different invariant objects

A degenerate Arnold diffusion mechanism in the Restricted 3 Body Problem

A major question in dynamical systems is to understand the mechanisms driving global instability in the 3 Body Problem (3BP), which models the motion of three bodies under Newtonian gravitational interaction. The 3BP is called restricted if one of the bodies has zero mass and the other two, the primaries, have strictly positive masses $m_0,m_1$. We consider the Restricted Planar Elliptic 3 Body Problem (RPE3BP) where the primaries revolve in Keplerian ellipses. We prove that the RPE3BP exhibits topological instability: for any values of the masses $m_0,m_1$ (except $m_0=m_1$), we build orbits along which the angular momentum of the massless body experiences an arbitrarily large variation provided the eccentricity of the orbit of the primaries is positive but small enough. In order to prove this result we show that a degenerate Arnold Diffusion Mechanism, which moreover involves exponentially small phenomena, takes place in the RPE3BP. Our work extends the result obtained in \cite{MR3927089} for the a priori unstable case $m_1/m_0\ll1$, to the case of arbitrary masses $m_0,m_1>0$, where the model displays features of the so-called \textit{a priori stable} setting