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### Open Access Journals

DCDS

We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations related to the birth of asymptotically stable, unstable and elliptic periodic orbits.

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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