The Gardner problem and cycle slipping bifurcation for type 2 phase-locked loops

Phase-locked loops (PLLs) are nonlinear control systems which are designed to synchronize a voltagecontrolled oscillator (VCO) signal with a reference one. PLLs have many applications in energy and robotic systems, satellite navigation, wireless and optical communications, cyber-physical systems [Du & Swamy, 2010; Karimi-Ghartemani, 2014; Rosenkranz & Schaefer, 2016; Best et al., 2016; Kaplan & Hegarty, 2017; Kuznetsov et al., 2020c; Zelenskii et al., 2021; Zelensky et al., 2021; Kuznetsov et al., 2022]. Analog PLLs can be described by systems of nonlinear differential equations with periodic right-hand sides, which are