66 research outputs found
Exponentially small splitting of separatrices beyond Melnikov analysis: rigorous results
We study the problem of exponentially small splitting of separatrices of one
degree of freedom classical Hamiltonian systems with a non-autonomous
perturbation fast and periodic in time. We provide a result valid for general
systems which are algebraic or trigonometric polynomials in the state
variables. It consists on obtaining a rigorous proof of the asymptotic formula
for the measure of the splitting. We obtain that the splitting has the
asymptotic behavior ,
identifying the constants in terms of the system features.
We consider several cases. In some cases, assuming the perturbation is small
enough, the values of coincide with the classical Melnikov approach.
We identify the limit size of the perturbation for which this theory holds
true. However for the limit cases, which appear naturally both in averaging and
bifurcation theories, we encounter that, generically, and are not
well predicted by Melnikov theory
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 . 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 (except ), 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 , to the case of arbitrary masses
, where the model displays features of the so-called \textit{a
priori stable} setting
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