Chaos from third-order phase-locked loops with a slowly varying parameter

The dynamic behavior of a third -order PLL (phase-locked loop) is studied by using a second-order loop filter for tracking frequency variable signals. The authors prove the existence of horseshoe chaos in the three-dimensional nonautonomous systems by the perturbation methods based on the ideas of Melnikov. This approach makes it possible to treat three-dimensional, periodically forced, slowly varying oscillators. The Lyapunov exponents and Lyapunov dimension are also calculated to confirm the theory. Theoretical results indicate that the parameter ranges where the chaos could occur are realistic in the typical designs. Computer simulations are performed to obtain the actual chaotic attractors