We study the dynamical behavior of the unstable periodic orbit (NHIM) associated to the non-return transition state (TS) of the H(2) + H collinear exchange reaction and their effects on the reaction probability. By means of the normal form of the Hamiltonian in the vicinity of the phase space saddle point, we obtain explicit expressions of the dynamical structures that rule the reaction. Taking advantage of the straightforward identification of the TS in normal form coordinates, we calculate the reaction probability as a function of the system energy in a more efficient way than the standard Monte Carlo method. The reaction probability values computed by both methods are not in agreement for high energies. We study by numerical continuation the bifurcations experienced by the NHIM as the energy increases. We find that the occurrence of new periodic orbits emanated from these bifurcations prevents the existence of a unique non-return TS, so that for high energies, the transition state theory cannot be longer applied to calculate the reaction probability.

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We investigate the classical dynamics of a hydrogen atom near a metallic surface in the presence of a uniform electric field. To describe the atom-surface interaction we use a simple electrostatic image model. Owing to the axial symmetry of the system, the z-component of the canonical angular momentum P is an integral and the electronic dynamics is modeled by a two degrees of freedom Hamiltonian in cylindrical coordinates. The structure and evolution of the phase space as a function of the electric field strength is explored extensively by means of numerical techniques of continuation of families of periodic orbits and Poincaré surfaces of section. We find that, due to the presence of the electric field, the atom is strongly polarized through two consecutive pitchfork bifurcations that strongly change the phase space structure. Finally, by means of the phase space transition state theory and the classical spectral theorem, the ionization dynamics of the atom is studied.

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We study the spin-up dynamics of a dual-spin spacecraft containing one axisymmetric rotor which is parallel to one of the principal axes of the spacecraft. It will be supposed that one of the moments of inertia of the platform is a periodic function of time and that the center of mass of the spacecraft is not modified. Under these assumptions, it is shown that in the absence of external torques and spinning rotors the system possesses chaotic behavior in the sense that it exhibits Smale's horseshoes. We prove this statement by means of the Melnikov method.The presence of chaotic behavior results in a random spin-up operation. This randomness is visualized by means of maps of the initial conditions with final nutation angle close to zero. This phenomenon is well described by a suitable parameter that measures the amount of randomness of the process. Finally, we relate this parameter with the Melnikov function in the absence of the spinning rotor and with the presence of subharmonic resonances.

The stability of the permanent rotations of a heavy gyrostat is analyzed by means of the Energy-Casimir method. Su cient and necessary conditions are established for some of the permanent rotations. The geometry of the gyrostat and the value of the gyrostatic moment are relevant in order to get stable permanent rotations. Moreover, the necessary conditions are also su cient, for some configurations of the gyrostat.

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