Homoclinic orbits in a simple chemical model of an autocatalytic reaction
In a one-parameter family of ODEs a periodic solution may disappear via the formation of a homoclinic orbit as the parameter c, say, passes through a critical value \(c^*\). The period of the solution becomes infinite, growing as \(\ln | c-c^*|\) as \(c\to c^*\) when the stationary solution at which the homoclinic orbit begins and ends is hyperbolic. If the stationary solution is non-hyperbolic, i.e. a bifurcation point, then the period tends to infinity more quickly than this logarithmic form, growing as \((| c-c^*|)^{-1/2}\). We show that although bifurcation point and homoclinic orbit formation are both codimension-1 phenomena, a homoclinic orbit related to a non- hyperbolic stationary solution is essentially a codimension-1 situation. Both types of homoclinic orbit have been found in a simple model for an autocatalytic chemical reaction scheme.
scientific article; zbMATH DE number 4077759