Persistence of instanton connections in chemical reactions with time-dependent rates

The evolution of a system of chemical reactions can be studied, in the eikonal approximation, by means of a Hamiltonian dynamical system. The fixed points of this dynamical system represent the different states in which the chemical system can be found, and the connections among them represent instantons or optimal paths linking these states. We study the relation between the phase portrait of the Hamiltonian system representing a set of chemical reactions with constant rates and the corresponding system when these rates vary in time. We show that the topology of the phase space is robust for small time-dependent perturbations in concrete examples and state general results when possible. This robustness allows us to apply some of the conclusions on the qualitative behavior of the autonomous system to the time-dependent situation.

Figure 1

Phase space for the branching and annihilating particle problem. The parameter values are $\sigma =\lambda =1$.

Figure 2

Sketch of the phase space for the branching and annihilating particle problem, assuming that the stable and unstable branches intersect. The heteroclinic connection goes through all the intersections of these branches.

Figure 3

Sketch of the phase space for the branching and annihilating particle problem, assuming that the stable and unstable branches do not intersect.

Figure 4

Sketch of the phase space for the branching and annihilating particle problem, assuming that the stable and unstable branches do not intersect.