Chaos and bifurcation in time delayed third order phase-locked loop

Abstract

In this paper, the modern nonlinear theory is applied to a third order phase locked loop (PLL) with a feedback time delay. Due to this delay, different behaviors that are not accounted for in a conventional PLL model are identified, namely, oscillatory instability, periodic doubling and chaos. Firstly, a Pade approximation is used to model the time delay where it is utilized in deriving the state space representation of the PLL under investigation. The PLL under consideration is simulated with and without time delay. It is shown that for certain loop gain (control parameter) and time delay values, the system changes its stability and becomes chaotic. Simulations show that the PLL with time delay becomes chaotic for control parameter value less than the one without time delay, i.e, the stable region becomes narrower. Moreover, the chaotic region becomes wider as time delay increases.