Transition to chaos

In the context of Hamiltonian dynamics I am interested in developing tools that can achieve an a priori determination of the regularity of a particular system. In this context, there exist a wide range of systems that undergo a transition from regular to chaotic behaviour as their energy is increased. The prediction of this energy threshold is very important for any potential application of the system under consideration.

Chaos transition for Hamiltonian systems [b.1,a.1,p3]
For the particular case of two-dimensional Hamiltonian systems subject to the potential $V(x,y)$ it is possible to formulate a criterion that predicts the energy threshold for the order-chaos transition [15,16]. The criterion originates from a linear stability analysis of the evolution of a small phase-space perturbation using Hamilton's equations. The geometrical interpretation of this criterion indicates that a necessary condition for chaotic evolution is that the orbit must reach a zone on the potential energy surface with negative Gaussian curvature $K(x,y)$ In fact simple Hamiltonian systems, such as elastic pendulums, rolling elastic cylinders (e.g. roller bearings), possess a region of positive Gaussian curvature near the minimum of the potential (rest position). Then, for low energy they are confined to regular behaviour inside a $K>0$ zone and as the energy is increased they may reach a zone with $K<0$. Thus, the energy threshold for the order-chaos transition may be estimated by the minimum energy necessary to reach a zone with negative Gaussian curvature on the potential energy surface.

Future directions: Transition to chaos for molecular vibrations
It is observed that CO$_2$ molecules possess a well defined, discrete, emission spectrum when exited at low energy. As the energy is increased above a certain threshold, the spectrum of the molecule widens indicating the signature of a more complex vibrational state. I am interested in applying the Gaussian curvature criterion (see above) to predict this energy threshold by modeling the CO$_2$ molecule using a classical mechanics approach. The simplest model is to consider the CO$_2$ molecule as two stiff torsional pendulums (one for each carbon-oxygen bond) and discard any oxygen-oxygen interaction. Within this approach, an estimation of the energy threshold could be obtained by applying the Gaussian curvature criterion to the dynamics of a single carbon-oxygen bond dynamics.