Extinction and intermittency of the chaotic attractor in phase-locked loop equation with periodic external forcing term

It is well-known that in the second-order non-autonomous nonlinear differential equation describing the periodically-forced phase-locked loop (PLL) circuit, the second-type (chaotic) attractor bifurcated from the second-type limit cycle which rotates the circumference of the cylindrical co-ordinate system, coexists with the first-type (chaotic) attractor bifurcated from the stable focus. Moreover, it is also known that there exists a merged chaotic attractor of these two attractors. In this research we obtain the regions of these (chaotic) attractors, and elucidate that the extinction mechanism of these attractors is mainly the heteroclinic tangency crisis, caused by the transition chain of the α- and the ω-branches of the inversely unstable fixed point (I) and the directly unstable fixed point (D). Moreover, we investigate the crisis induced intermittency observed when the first-type and the second-type chaotic attractors are merged together due to an explosion of attractors