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DCDS

We consider an example of singular or weakly hyperbolic Hamiltonian, with
$3$ degrees of freedom, as a model for the behaviour of a nearly-integrable
Hamiltonian near a simple resonance. The model consists of an integrable
Hamiltonian possessing a $2$-dimensional hyperbolic invariant torus with fast
frequencies $\omega/\sqrt\varepsilon$ and coincident whiskers, plus a
perturbation of order $\mu=\varepsilon^p$. We choose $\omega$ as the golden
vector. Our aim is to obtain asymptotic estimates for the splitting,
proving the existence of transverse intersections between the perturbed
whiskers for $\varepsilon$ small enough, by applying the Poincaré-Melnikov
method together with a accurate control of the size of the error term.

The good arithmetic properties of the golden vector allow us to prove that the splitting function has 4 simple zeros (corresponding to nondegenerate critical points of the splitting potential), giving rise to 4 transverse homoclinic orbits. More precisely, we show that a shift of these orbits occurs when $\varepsilon$ goes across some critical values, but we establish the continuation (without bifurcations) of the 4 transverse homoclinic orbits for all values of $\varepsilon\to0$.

The good arithmetic properties of the golden vector allow us to prove that the splitting function has 4 simple zeros (corresponding to nondegenerate critical points of the splitting potential), giving rise to 4 transverse homoclinic orbits. More precisely, we show that a shift of these orbits occurs when $\varepsilon$ goes across some critical values, but we establish the continuation (without bifurcations) of the 4 transverse homoclinic orbits for all values of $\varepsilon\to0$.

DCDS

We consider a singular or weakly hyperbolic Hamiltonian, with $n+1$ degrees of
freedom, as a model for the behaviour of a nearly-integrable Hamiltonian near
a simple resonance. The model consists of an integrable Hamiltonian possessing
an $n$-dimensional hyperbolic invariant torus with fast frequencies
$\omega/\sqrt\varepsilon$ and coincident whiskers, plus a perturbation of order
$\mu=\varepsilon^p$. The vector $\omega$ is assumed to satisfy a Diophantine
condition.

We provide a tool to study, in this singular case, the splitting of the perturbed whiskers for $\varepsilon$ small enough, as well as their homoclinic intersections, using the Poincaré--Melnikov method. Due to the exponential smallness of the Melnikov function, the size of the error term has to be carefully controlled. So we introduce flow-box coordinates in order to take advantage of the quasiperiodicity properties of the splitting. As a direct application of this approach, we obtain quite general upper bounds for the splitting.

We provide a tool to study, in this singular case, the splitting of the perturbed whiskers for $\varepsilon$ small enough, as well as their homoclinic intersections, using the Poincaré--Melnikov method. Due to the exponential smallness of the Melnikov function, the size of the error term has to be carefully controlled. So we introduce flow-box coordinates in order to take advantage of the quasiperiodicity properties of the splitting. As a direct application of this approach, we obtain quite general upper bounds for the splitting.

ERA-MS

We study the splitting of invariant manifolds of whiskered tori with two or
three frequencies in nearly-integrable Hamiltonian systems,
such that the hyperbolic part is given by a pendulum.
We consider a 2-dimensional torus with
a frequency vector $\omega=(1,\Omega)$, where $\Omega$ is a quadratic
irrational number, or a 3-dimensional torus with a frequency vector
$\omega=(1,\Omega,\Omega^2)$, where $\Omega$ is a cubic irrational number.
Applying the Poincaré--Melnikov method, we find exponentially small
asymptotic estimates for the maximal splitting distance between the stable and
unstable manifolds associated to the invariant torus, and we show that such
estimates depend strongly on the arithmetic properties of the frequencies. In
the quadratic case, we use the continued fractions theory to establish a
certain arithmetic property, fulfilled in 24 cases, which allows us to provide
asymptotic estimates in a simple way. In the cubic case, we focus our attention
to the case in which $\Omega$ is the so-called cubic golden number (the real
root of $x^3+x-1=0$), obtaining also asymptotic estimates. We point out the
similitudes and differences between the results obtained for both the quadratic
and cubic cases.

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