Fractal Oscillators

Abstract

We consider a weakly nonlinear oscillator with a fractal forcing, given by the Weierstrass function, and use the asymptotic perturbation (AP) method to study its behavior.  Being this function nowhere differentiable we can only use adequate approximations. We find that while in the linear case the resulting motion is a simple superposition between the fractal forcing and the standard oscillation, on the contrary in the nonlinear case the oscillator phase and its frequency also become fractal. We obtain the Poincarè sections in various cases and all theoretical findings are corroborated with numerical simulation.