The first mathematical problems of the global analysis of dynamical models can be traced back to the engineering problem of the Watt governor design. Engineering requirements and corresponding mathematical problems led to the fundamental discoveries in the global stability theory. Boundaries of global stability in the space of parameters are limited by the birth of oscillations. The excitation of oscillations from unstable equilibria can be easily analysed, while the revealing of oscillations not connected with equilibria is a challenging task being studied in the theory of hidden oscillations. In this survey, a brief history of the first global stability criteria development and corresponding counterexamples with hidden oscillations are discussed.

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In the present work, a second-order type-2 phase-locked loop (PLL) with a piecewise-linear phase detector characteristic is analyzed. An exact solution to the Gardner problem on the lock-in range is obtained for the considered model. The solution is based on a study of cycle slipping bifurcation, which improves the well-known engineering estimates.