Rate of convergence of numerical approximations to homoclinic bifurcation points

Abstract

For x = f(x, λ), x ∈ R n , λ ∈ R, having a hyperbolic or semihyperbolic equilibrium p(λ), we study the numerical approximation of parameter values λ* at which there is an orbit homoclinic to p(λ). We approximate λ* 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 approximations to appropriate invariant manifolds of p(λ). A phase condition is also necessary to make the solution unique. Using a lemma of Xiao-Biao Lin, we improve, for certain phase conditions, existing estimates on the rate of convergence of the computed homoclinic bifurcation parameter value λ, to the true λ* value λ*. The estimates we obtain agree with the rates of convergence observed in numerical experiments. Unfortunately, the phase condition most commonly used in numerical work is not covered by our results