In this article some qualitative and geometric aspects of non-smooth dynamical systems theory are discussed. The main aim of\ud
this article is to develop a systematic method for studying local(and global) bifurcations in non-smooth dynamical systems. Our results deal with the classification and characterization of generic codimension-2 singularities of planar Filippov Systems as well as the presentation of the bifurcation diagrams and some dynamical consequencesPeer ReviewedPostprint (published version
Fix s > 1. Colliander, Keel, Staffilani, Tao and Takaoka proved in [CKS + 10] the existence of solutions of the cubic defocusing nonlinear Schrödinger equation in the two torus with s-Sobolev norm growing in time. In this paper we generalize their result to the cubic defocusing nonlinear Schrödinger equation with a convolution potential. Moreover, we show that the speed of growth is the same as the one obtained for the cubic defocusing nonlinear Schrödinger equation in [GK12]. The results we obtain can deal with any potential in H s0 (T 2 ), s 0 > 0.Are there solutions of (5) with periodic boundary conditions in dimension 2 or higher with unbounded growth of H s -norm for s > 1?Moreover, he conjectured, that in case this is true, the upper bound that he had obtained in [Bou96] was not optimal and that the growth should be subpolynomial in time, that is, u(t) H s ≪ t ε u(0) H s for t → ∞, for all ε > 0.
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 ε β e −a/ε , 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.
We consider the completely resonant defocusing non-linear Schrödinger equation on the two dimensional torus with any analytic gauge invariant nonlinearity. Fix s > 1. We show the existence of solutions of this equation which achieve arbitrarily large growth of H s Sobolev norms. We also give estimates for the time required to attain this growth.
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