A continuation algorithm for discovering new chaotic motions in forced Duffing systems

We study a dynamical system described by the Duffing equation under static plus large periodic excitation. The bifurcation diagram in terms of the amplitude of the periodic excitation reveals that there are at least six cascades of period doubling bifurcations with the other parameters fixed at specific values. These six sequences are further investigated by a continuation algorithm which is based on the principles of the shooting method combined with the Newton method for solving nonlinear equations. A Runge-Kutta-Hûta method has been used for solving the system of differential equations. The final conclusion is that each of the six sequences is governed by Feigenbaumʹs number δ = 4.6692 from Universality Theory. Beyond the limit values derived for the amplitude of the periodic excitation new strange attractors are found.