Application of high-order difference methods for the study of period doubling bifurcations in nonlinear oscillators

Abstract

For the study of period doubling bifurcations in nonlinear T-periodic forced oscillators depending on one parameter, it is necessary to use a quick and accurate method to obtain periodic solutions of a second-order nonlinear differential equation. Using a periodic approximation, obtained earlier for another parametric value, a Newton method allows its iterative refinement, by the successive solution of linear periodic boundary value problems. In these problems, the derivatives of the unknown periodic solution are approximated by high-order difference formulae, yielding accurate results in the N abscissae x i = iT N , i = 0, …, N − 1. To determine the parametric values where bifurcation takes place, the solution of the variational equations was calculated using a Runge-Kutta-Hǔta method. Therefore the values of the corresponding nT-periodic solution ( n = 2 k , k ∈ N ) were determined in some points inside the discretization intervals using trigonometric interpolation. From the resulting fundamental matrix Φ( nT), the multipliers are readily determined, allowing the detection of the period doubling bifurcations when the largest multiplier passes through −1. The method has been applied successfully to study bifurcations leading to chaos in several classical nonlinear oscillators.