Bifurcations of dividing surfaces in chemical reactions.

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.

Transition state theory for laser-driven reactions.

Recent developments in transition state theory brought about by dynamical systems theory are extended to time-dependent systems such as laser-driven reactions. Using time-dependent normal form theory, the authors construct a reaction coordinate with regular dynamics inside the transition region. The conservation of the associated action enables one to extract time-dependent invariant manifolds that act as separatrices between reactive and nonreactive trajectories and thus make it possible to predict the ultimate fate of a trajectory. They illustrate the power of our approach on a driven Henon-Heiles system, which serves as a simple example of a reactive system with several open channels. The present generalization of transition state theory to driven systems will allow one to study processes such as the control of chemical reactions through laser pulses.

Dissipative-Hamiltonian decomposition of smooth vector fields based on symmetries.

We propose a method to decompose a smooth vector field into conservative and dissipative components. The procedure is based on the identification of the kernel of a linear operator associated with a given Hamiltonian combined with the use of Lie transformations for vector fields. Moreover, under certain conditions the nonconservative part of the splitting can be dropped at a given order of the transformation, obtaining after truncation, a Hamilton vector field. The technique is illustrated through the application to the motion of a particle subject to the potential of a champagne bottle plus a small friction.

Invariant manifolds of an autonomous ordinary differential equation from its generalized normal forms.

A method to approximate some invariant sets of dynamical systems defined through an autonomous m-dimensional ordinary differential equation is presented. Our technique is based on the calculation of formal symmetries and generalized normal forms associated with the system of equations, making use of Lie transformations for smooth vector fields. Once a symmetry is determined up to a certain order, a reduction map allows us to pass from the equation in normal form to a related equation in a certain reduced space, the so-called reduced system of dimension s