Complex orbits in a second-order digital filter with sinusoidal response

This paper analyzes a two-dimensional piecewise affine map driven by a periodic perturbation, which is relevant to the second-order digital filters with sinusoidal response. By using symbolic dynamics method, a formula for an arbitrary iterate of the map is derived. When overflow occurs, the system is nonlinear. If the corresponding symbolic sequences are aperiodic, some orbits are computed and illustrated. These show that the orbit structure is much richer than that of the autonomous and step-response cases. And numerical experiments to estimate the fractal dimension of the chaotic cases are performed with fractal Brownian motion. Finally, higher order periodic trajectories are found via the particle swarm optimization.