Methods for locating periodic orbits in highly unstable systems

Computing periodic orbits with arbitrary precision

This paper deals with the computation of periodic orbits of dynamical systems up to any arbitrary precision. These very high requirements are useful, for example, in the studies of complex pole location in many physical systems. The algorithm is based on an optimized shooting method combined with a numerical ordinary differential equation (ODE) solver, tides, that uses a Taylor-series method. Nowadays, this methodology is the only one capable of reaching precision up to thousands of digits for ODEs. The method is shown to be quadratically convergent. Some numerical tests for the paradigmatic Lorenz model and the Hénon-Heiles Hamiltonian are presented, giving periodic orbits up to 1000 digits.

Analysis of the HO2 Vibrational Spectrum on an Accurate Ab Initio Potential Energy Surface

The complete vibrational spectrum of the HO2(X(2)A' ') radical, up to the H + O2 dissociation limit, has been determined quantum mechanically on an accurate potential energy surface (PES), based on approximately 15000 ab initio points at the icMRCI+Q/aug-cc-pVQZ level of theory. The vibrational states are found to be assignable at low energies but become more irregular as the energy approaches the dissociation limit. However, even at very high energies, regularity still exists, in sharp contrast to earlier results based on the double many-body expansion (DMBE) IV potential. Several Fermi resonances have been identified, and the spectrum is fit with a spectroscopic Hamiltonian. In addition, the vibrational dynamics is analyzed using a periodic orbit approach.

Periodic orbits in biological molecules: Phase space structures and selectivity in alanine dipeptide

Small and large molecules may localize their energy in specific bonds or generally in vibrational modes for extended periods of time, an effect which may have dramatic consequences in reaction dynamics. Periodic orbits offer the means to identify phase space regions with localized motions. The author demonstrate that techniques to locate periodic orbits developed for small molecules can be applied to large molecules such as alanine dipeptide. The widely used empirical force fields are employed and principal families of periodic orbits associated with local-type motions and emanated from the lowest energy minima and saddle points are investigated. Continuation of these families at high energies unravels the stable and unstable regions of phase space as well as elementary bifurcations such as saddle nodes.