Shilnikov orbits in an autonomous third-order chaotic phase-locked loop

In this work we investigate the Shilnikov homoclinic bifurcation in a new type of phase-locked loop (PLL) having a second-order loop filter. This system can be represented as a third-order autonomous system with piecewise-linear characteristics. By using piecewise-linear analysis, bifurcation equations for many types of homoclinic orbits are derived. Solving these equations gives many Shilnikov-type homoclinic orbits. We present bifurcation diagrams for the homoclinic orbits in the gain (K₀) versus detuning (Δω) plane. Finally, we demonstrate the role of the homoclinic orbits in the global bifurcation of attractors both by computer simulation and experiments,