Determination of invariant tori bifurcation points

Two numerical methods are presented for the determination of invariant tori bifurcation points on a branch of periodic solutions of ordinary differential equations (lumped parameter systems). The methods are based on the shooting technique and the Newton method. The coefficients of characteristic polynomial of the monodromy matrix have to be evaluated in each iteration, however, a modification is suggested where it is not necessary. The convergence of the methods is very sensitive to an initial guess. A good guess can be obtained from results of continuation of the periodic solutions on a parameter. Applications to a hydrodynamic problem (a seven-mode model truncated Navier-Stokes equations for a two-dimensional flow of an incompressible fluid) and a problem from chemical reactor theory (two well mixed reaction cells with linear diffusion coupling and the Brusselator reaction kinetic scheme) demonstrate the effectiveness of the methods.