| Wang Lian,Wang Muqiu.[J].数学年刊A辑,1980,1(3-4):411~419 |
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| QUALITATIVE METHOD IN ANALYSIS OF A P-L-LWITH TANGENT CHARACTERISTIC ANDFREQUENCY MODULATION INPUT |
| Received:September 29, 1979 Revised:December 29, 1979 |
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| In analysis of p-L-L with tangent characteristic and frequency modulation input, we have obtained the following two types of the phase looked loop equation.
\[\begin{array}{l}
\frac{{{\partial ^2}\varphi }}{{\partial {t^2}}} + \alpha \frac{{d\varphi }}{{dt}} + \gamma \tan \varphi = {\beta _1} + {\beta _2}(\cos {\Omega _M}t + {\Omega _M}\sin {\Omega _M}t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (I)\\frac{{{\partial ^2}\varphi }}{{\partial {t^2}}} + (\alpha + \eta {\sec ^2}\varphi )\frac{{d\varphi }}{{dt}} + \gamma \tan \varphi = {\beta _1} + {\beta _2}(\cos {\Omega _M}t - {\Omega _M}\sin {\Omega _M}t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (II) \(\alpha > 0,\gamma > 0,\eta > 0,{\beta _1} > 0,{\beta _2} > 0,{\Omega _M} > 0)
\end{array}\]
In this paper, our aim is to explain the usual qualitative method and Lyapunov's function method, by which the existence of a periodic solution of (I), (II) is established. In addition, we especially point out: How is to construct the Lyapunovas function
for the nonlinear and nonairtoiiomous system? This is a very important problem. |
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