Dynamics on Manifolds

In Section 1, the local analysis of the stability properties of nonlinear equations was discussed. In the 1880s Poincare discovered a theorem that makes a statement about the entire set of critical points of dynamical systems on two-dimensional manifolds. This result was later extended by Hopf to arbitrary manifolds (see Appendix for greater detail and references). Here the theorem is applied to dynamics in chemical systems. 

By examining the physical constraints imposed on kinetic equations, it is possible to embed many chemical reaction systems on N—spheres. Assume that the reaction kinetics are given by Eqs. (1) and (2), where each of the N reactants is present initially and no new reactants are generated as time proceeds. Since chemical reactions are reversible, it is impossible for any reactant to disappear completely. Consequently, we assume that there exists a small number 8 such that 

since the concentrations must remain finite we assume that, for some large number C, 

Boundary conditions (12) and (13) constrain the dynamics to a box in N dimensions, where all trajectories on the boundary of the box enter the box. By associating the boundary of the box with a single unstable source at the South Pole, the chemical kinetic system defined by (1), (12), and (13) can be embedded on an N-sphere. By applying (9)-(ll) we compute 

Equation (14) places sharp restrictions on the sets of critical points in chemical reaction networks. Consider the simplest cases that are possible for two reacting chemicals, N-2. 

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