Numerical integration of stiff ODE's of singular perturbation type (Q1175297)

A numerical procedure is presented to approximate solutions to the singularly perturbed system (i) \(x'=f(x,y,\varepsilon)\), \(\varepsilon y'=g(x,y,\varepsilon)\), \(x\in R^ m\), \(y\in R^ n\), which is based on known behavior of the trajectories of the system. Conditions are placed on \(f\), \(g\) such that for small \(\varepsilon\) there exists a smooth invariant attractive manifold \(M\) of trajectories which is described by \(y=s(x,\varepsilon)\). To approximate solutions \(x(t)\), \(y(t)\) of (i) with \(\varepsilon\) fixed and \(x(t_ 0)=x_ 0\), \(y(t_ 0)=y_ 0\), \((x_ 0,y_ 0)\in M\), the author generates approximations \(S(x,\varepsilon)\) to \(s(x,\varepsilon)\) and solves (ii) \(x'=f(x,S(x,\varepsilon),\varepsilon)\), \(x(t_ 0)=x_ 0\), numerically to obtain \(X(t)\). The pair \(X(t)\), \(Y(t)\), \(Y(t)=S(X(t),\varepsilon)\) is used to approximate \(x(t)\), \(y(t)\) on \(M\). When the initial values of a trajectory \(x_ 0\), \(y_ 0\) are not on \(M\) then the system (i) is first approximately solved numerically using a fast time scale until the manifold \(M\) is reached and then the above procedure is applied. An analysis of the order of the error in the computation is presented. Details of a Fortran code written to implement the numerical procedure are given and numerical results presented for a system introduced by \textit{E. C. Zeeman} [Dynamical Syst., Proc. Sympos. Univ. Bania, Salvador 1971, 683-741 (1973; Zbl 0289.92004)] which models the local behavior of a nerve impulse.