Existence uniqueness and asymptotic stability of periodic solutions of a nonlinear equation in phase locked technology (Q1179417)

Formerly an equation was analysed, which had a ``tangent detected phase'' character, and there was shown why the phase locked loop has not lock/losing point, but in that time according to practical experts' opinion it was difficult to put such a phase locked loop into practice. Later \textit{W. C. Lindsey} [Synchronization system in communication and control, New Jersey (1972)] constructed such a circuit with the character of detected phase, for which \(g(\varphi)=(1+k)\sin \varphi\cdot[1+k \cos \varphi]^{-1}\), \(0\leq k<1\). Accordingly, the subject of investigation is the equation \[ {d^ 2\varphi \over dt^ 2}+f\left(\varphi,{d\varphi\over dt}\right) {d\varphi \over dt}+g(\varphi)=\mu e(t), \] where \(\mu>0\); \(f\) is continuous and differentiable; \(e\) is continuous. Existence and uniqueness of the periodic solution is proved on the cylinder \(-\pi\leq\varphi\leq\pi\); \(- \infty<{d\varphi \over dt}<+\infty\). Conditions for asymptotic stability of the solution are shown.