The computation of paths of homoclinic orbits

The dynamical analysis of systems of nonlinear, autonomous, ordinary differential equations (ODEs), of the general form ${d\vec y\over dt}+\vec f(\vec y, \vec\lambda)=\vec 0,$ typically involves studying the dependence of solutions to the system of ODEs on the parameter $\vec\lambda;$ qualitatively different types of solutions, which are of both theoretical and practical interest, include steady state, periodic, homoclinic and heteroclinic. A non-constant solution $\vec y(t)$ for $t\in (-\infty,+\infty),$ of ${d\vec y\over dt}+\vec f(\vec y,\vec\lambda)=\vec 0$ at some $\vec\lambda,$ is called a homoclinic orbit if $lim\sb{t\to{\pm}\infty}\vec y(y)=\vec y\sb c$ where $\vec y\sb c$ is a steady state (i.e., $\vec f(\vec y\sb c,\vec\lambda)=\vec0).$ The computation of paths of homoclinic orbits is important for the complete dynamical analysis of a system of ODEs. In particular, paths of homoclinic orbits separate regions in parameter space for which periodic solutions exist, and for which there are no periodic solutions. A stationary point of ${d\vec y\over dt}+\vec f(\vec y,\vec\lambda)=\vec0$ is a point ($\vec y\sb c,\vec\lambda)$ that satisfies $\vec f(\vec y\sb c,\vec\lambda)=\vec0.$ Existing methods for the numerical computation of paths of homoclinic orbits at hyperbolic steady states suffer from several drawbacks, particularly if the stationary point is near a point on a fold. These drawbacks are identified. We present a new method for the numerical computation of paths of homoclinic orbits at hyperbolic steady states, which avoids the drawbacks of current homoclinic orbit methods. Current methods for the computation of paths of homoclinic orbits treat only the case where the steady state is hyperbolic. We present a new method for the numerical computation of paths of homoclinic orbits where the stationary point is a point on a fold; this appears to be the first effective method for such a problem. Our homoclinic orbit methods (i.e., at hyperbolic steady states, and on a fold) are applied to compute paths of homoclinic orbits of two practical systems of ODEs, which model certain chemical reactions. Bifurcation diagrams, which include projections of computed paths of homoclinic orbit solutions to these systems of ODEs, are presented.