arXiv subjects

Marcel Guardia

Publications and source records attributed to Marcel Guardia.

41 records · Page 3Linked to original sources

arXiv2012

Oscillatory motions for the restricted planar circular three body problem

In this paper we consider the circular restricted three body problem which models the motion of a masless body under the influence of the Newtionan gravitational force caused by two other bodies, the primaries, which move along cicular planar Keplerian orbits. In a suitable system of coordinates, this system has two degrees of freedom and the conserved energy is usually called the Jacobi constant. In 1980, J. Llibre and C. Simó proved the existence of oscillatory motions for the restricted planar circular three body problem, that is, of orbits which leave every bounded region but which return infinitely often to some fixed bounded region. To prove their existence they had to assume the ratio between the masses of the two primaries to be exponentially small with respect to the Jacobi constant. In the present work, we generalize their work proving the existence of oscillatory motions for any value of the mass ratio. To obtain such motions, we show that, for any value of the mass ratio and for big values of the Jacobi constant, there exist transversal intersections between the stable and unstable manifolds of infinity which guarantee the existence of a symbolic dynamics that creates the oscillatory orbits. The main achievement is to rigorously prove the existence of these orbits without assuming the mass ratio small since then this transversality can not be checked by means of classical perturbation theory respect to the mass ratio. Since our method is valid for all values of mass ratio, we are able to detect a curve in the parameter space, formed by the mass ratio and the Jacobi constant, where cubic homoclinic tangencies between the invariant manifolds of infinity appear.

math.DS↗

arXiv2012

Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix $s>1$. Colliander, Keel, Staffilani, Tao and Takaoka proved in \cite{CollianderKSTT10} the existence of solutions with $s$-Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is $c>0$ such that for any $\mathcal{K}\gg 1$ we find a solution $u$ and a time $T$ such that $\| u(T)\|_{H^s}\geq\mathcal{K} \| u(0)\|_{H^s}$. Moreover, time $T$ satisfies polynomial bound $0<T<\mathcal{K}^c$.

math.AP↗

arXiv2012

Splitting of separatrices in the resonances of nearly integrable Hamiltonian Systems of one and a half degrees of freedom

In this paper we consider general nearly integrable analytic Hamiltonian systems of one and a half degrees of freedom which are a trigonometric polynomial in the angular state variable. In the resonances of these systems generically appear hyperbolic periodic orbits. We study the possible transversal intersections of their invariant manifolds, which is exponentially small, and we give an asymptotic formula for the measure of the splitting. We see that its asymptotic first order is of the form $K \varepsilon^β \text{e}^{-a/\varepsilon}$ and we identify the constants $K,β,a$ in terms of the system features. We compare our results with the classical Melnikov Theory and we show that, tipically, in the resonances of nearly integrable systems Melnikov Theory fails to predict correctly the constants $K$ and $β$ involved in the formula.

math.DS↗

arXiv2012

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 $K \varepsilon^β \text{e}^{-a/\varepsilon}$, identifying the constants $K,β,a$ in terms of the system features. We consider several cases. In some cases, assuming the perturbation is small enough, the values of $K,β$ 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, $K$ and $β$ are not well predicted by Melnikov theory.

math.DS↗

arXiv2011

Exponentially and non-exponentially small splitting of separatrices for the pendulum with a fast meromorphic perturbation

In this paper we study the splitting of separatrices phenomenon which arises when one considers a Hamiltonian System of one degree of freedom with a fast periodic or quasiperiodic and meromorphic in the state variables perturbation. The obtained results are different from the previous ones in the literature, which mainly assume algebraic or trigonometric polynomial dependence on the state variables. As a model, we consider the pendulum equation with several meromorphic perturbations and we show the sensitivity of the size of the splitting on the width of the analyticity strip of the perturbation with respect to the state variables. We show that the size of the splitting is exponentially small if the strip of analyticity is wide enough. Furthermore, we see that the splitting grows as the width of the analyticity strip shrinks, even becoming non-exponentially small for very narrow strips. Our results prevent from using polynomial truncations of the meromorphic perturbation to compute the size of the splitting of separatrices.

math.DS↗