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For x=f (x, ), x Rn, R, having a hyperbolic or semihyperbolicequilibrium p(), we study the numerical approximation of parametervalues * at which there is an orbit homoclinic to p(). We approximate* by solving a finite-interval boundary value problem on J=[T,T+], T<0<T+, with boundary conditions that sayx(T) and x(T+) are in approximations to appropriate invariantmanifolds of p(). A phase condition is also necessary to makethe solution unique. Using a lemma of Xiao-Biao Lin, we improve,for certain phase conditions, existing estimates on the rateof convergence of the computed homoclinic bifurcation parametervalue , to the true value *. The estimates we obtain agree withthe rates of convergence observed in numerical experiments.Unfortunately, the phase condition most commonly used in numericalwork is not covered by our results. 相似文献
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