RATE OF CONVERGENCE OF NUMERICAL APPROXIMATIONS TO HOMOCLINIC BIFURCATION POINTS

For x = f(x, lambda), x is an element of R(n), lambda is an element of R, having a hyperbolic or semihyperbolic equilibrium p(lambda), we st udy the numerical approximation of parameter values lambda at which t here is an orbit homoclinic to p(A). We approximate lambda by solving a finite-interval boundary value problem on J = [T--, T-+], T-- < 0 < T-+, with boundary conditions that say x(T--) and x(T-+) are in appro ximations to appropriate invariant manifolds of p(lambda). A phase con dition is also necessary to make the solution unique. Using a lemma of Xiao-Biao Lin, we improve, for certain phase conditions, existing est imates on the rate of convergence of the computed homoclinic bifurcati on parameter value lambda(j) to the true value lambda. The estimates we obtain agree with the rates of convergence observed in numerical ex periments. Unfortunately, the phase condition most commonly used in nu merical work is not covered by our results.