Rate of convergence of numerical approximations to homoclinic bifurcation points

Abstract

For x= f ( x , λ), x ε R n , λ ε 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 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.