UPOsHam: A Python package for computing unstable periodic orbits in two degrees of freedom Hamiltonian systems

Software repository

Statement of Need

In Hamiltonian dynamical systems the fundamental phase space structure that

partitions dynamically disparate trajectories and mediates transition between multi-stable regions is an invariant manifold. These are anchored to the normally hyperbolic invariant manifold in a general N degree of freedom system and this becomes an unstable periodic orbit in the two degree of freedom case. This Python package provides a collection of three different methods for computing the unstable periodic orbits (UPOs) at a specified total energy. Even though, there is no lack of numerical methods for this computing UPOs, we have found they either lack in reproducibility or written using closed source software. Our aim is to provide an open source package that implements the three methods and shows examples for applied scientists.

Summary

UPOsHam is a collection of three useful methods for computing unstable periodic orbits in Hamiltonian systems that model a diverse array of problems in physical sciences and engineering. The methods have been implemented for three example Hamiltonian systems that are prototypical models of chemical reactions.

We show few example computations of the unstable periodic orbit for these examples in Fig.

Features

In this package, the user has the option to choose between the three methods described below. These are implemented in separate scripts with functions that define the model specific quantities.

Turning point

Hamiltonian systems of the form kinetic plus potential energy have unstable periodic orbits in the bottleneck that touches the equipotential lines given by \(V(x,y) = E\). This method is based on finding the UPO by checking for trajectories that turn in the opposite directions and iteratively bringing them closer to approximate the UPO (Pollak, Child, and Pechukas 1980).

Differential turning point

Differential correction and numerical continuation

Examples

Consider the following two degrees-of-freedom Hamiltonian model of a reaction in a bath (solvent)

Reaction and bath modes uncoupled

\[\begin{equation} \mathcal{H}(x,y,p_x,p_y) = \frac{p_x^2}{2} - \alpha \frac{x^2}{2} + \beta \frac{x^4}{4} + \frac{\omega}{2}\left( p_y^2 + y^2 \right) \end{equation}\]

Reaction and bath modes coupled

\[\begin{equation} \mathcal{H}(x,y,p_x,p_y) = \frac{p_x^2}{2} - \alpha \frac{x^2}{2} + \beta \frac{x^4}{4} + \frac{\omega}{2}\left( p_y^2 + y^2 \right) + \frac{\epsilon}{2}(x - y)^2 \end{equation}\]

DeLeon-Berne model of unimolecular isomerization

This Hamiltonian is a clasic example that models the unimolecular conformational isomerization and applied dynamical systems tools to study chemical reactions(Nelson De Leon and Berne 1981; N De Leon and Marston 1989).

\[\begin{equation} \mathcal{H}(x,y,p_x,p_y) = T(p_x, p_y) + V_{\rm DB}(x, y) = \frac{p_x^2}{2m_A} + \frac{p_y^2}{2m_B} + V_{\rm DB}(x, y) \end{equation}\]

where the potential energy function \(V_{\rm DB}(x,y)\) is

\[\begin{equation} \begin{aligned} V_{\rm DB}(x,y) = & V(x) + V(y) + V(x,y) \\ V(y) = & \dfrac{\mathcal{V}^{\ddagger}}{y_w^4}y^2(y^2 - 2y_w^2) + \epsilon_s \\ V(x) = & D_x\left[ 1 - \exp(-\lambda x) \right]^2 \\ V(x,y) = & \dfrac{\mathcal{V}^{\ddagger}}{y_w^4}y^2(y^2 - 2y_w^2)\left[ \exp(-\zeta \lambda x) - 1 \right] \end{aligned} \label{eqn:pot_energy_db} \end{equation}\]

The parameters in the model are \(m_A, m_B\) which represent mass of the isomers and is constant $ m_A = m_B = $, while \(\epsilon_s, D_x, y_w, \mathcal{V}^{\ddagger}\) denote the energy of the saddle, dissociation energy of the Morse oscillator, location of the isomerization wells, potential energy of the saddle equilibrium point, respectively, and will be kept fixed in this study, \(\lambda, \zeta\) denote the range of the Morse oscillator and coupling parameter between the \(x\) and \(y\) configuration space coordinates, respectively.

Visualization: Unstable periodic orbits in the bottleneck

Uncoupled system

Coupled system

De Leon-Berne system

Relation to ongoing research projects

We are developing geometric methods of phase space transport in the context of chemical reaction dynamics that rely heavily on identifying and computing the unstable periodic orbits. Manuscript related to the De Leon-Berne model is under preparation.

Acknowledgements

We acknowledge the support of EPSRC Grant No. EP/P021123/1 and Office of Naval Research (Grant No. N00014-01-1-0769). The authors would like to acknowledge the London Mathematical Society and School of Mathematics at University of Bristol for supporting the undergraduate research bursary. We acknowledge contributions from Shane Ross for writing the early MATLAB version of the differential correction and numerical continuation code.

References

De Leon, N, and C. Clay Marston. 1989. “Order in Chaos and the Dynamics and Kinetics of Unimolecular Conformational Isomerization.” The Journal of Chemical Physics 91 (6): 3405–25. doi:10.1063/1.456915.

De Leon, Nelson, and B. J. Berne. 1981. “Intramolecular Rate Process: Isomerization Dynamics and the Transition to Chaos.” The Journal of Chemical Physics 75 (7): 3495–3510. doi:10.1063/1.442459.

Pollak, Eli, Mark S. Child, and Philip Pechukas. 1980. “Classical Transition State Theory: A Lower Bound to the Reaction Probability.” The Journal of Chemical Physics 72 (3): 1669–78. doi:10.1063/1.439276.