A convergence-improving iterative method for computing periodic orbits near Bifurcation Points

The accurate computation of periodic orbits and the precise knowledge of their bifurcation properties are very important for studying the behavior of many dynamical systems of physical interest. In this paper, we present an iterative method for computing periodic orbits, which has the advantage of improving the convergence of previous Newton-like schemes, especially near bifurcation points. This method is illustrated here on a conservative, nonlinear Mathieu equation, for which a sequence of period-doubling bifurcations is followed, long enough to obtain accurate estimates of the two universal scaling constants α, β, as well as the universal rate δ, by which the bifurcation values of a parameter q = qk, k =1, 2, 3, ..., tend to their limiting value, q_∞ < ∞, as k increases.