CHAOS FROM 3RD-ORDER PHASE-LOCKED LOOPS WITH A SLOWLY VARYING PARAMETER

A phase-locked loop (PIX) used in FM demodulator has been proven by Endo and Chua to have the chaotic phenomenon via the Melnikov method. However, the loop filter used in that system is a first-order lead-lag type. The limitation of the resultant second-order PLL is that the loop will not lock if the received signal frequency exceeds an upper bound. Hence, in this paper, we will study the dynamic behavior of a third-order PLL by using a second-order loop filter for tracking frequency-variable signals. We prove the existence of horseshoe chaos in the three-dimensional nonautonomous systems by the perturbation methods based on the ideas of Melnikov. This method permits us to treat three-dimensional periodically forced slowly varying oscillators. Moreover, 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. Finally, computer simulations are performed to obtain the actual chaotic attractors. © 1990 IEEE