A note on periodic solutions of a differential equation with two parameters |
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Authors: | Hsin Chu |
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Institution: | 1. University of Maryland, 20742, College Park, MD, USA
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Abstract: | In 1] (p. 215), the authors Andronov, Leontovich-Andronova, Gordon, and Maier, consider the following equation: $$\left\{ \begin{gathered} \tfrac{{dx}}{{dt}} = y, \hfill \\ \tfrac{{dy}}{{dt}} = x + x^2 - \left( {\varepsilon _1 + \varepsilon _2 x} \right)y, \hfill \\ \end{gathered} \right.$$ whereε 1 andε 2 are real constants andε 1 andε 2 are not both zero. They proved that there are no non-trivial periodic solutions except possibly for the case $0< \tfrac{{\varepsilon _1 }}{{\varepsilon _2 }}< \tfrac{3}{2}$ . They left that case as an open problem. In this note we prove that there are indeed no non-trivial periodic solutions in the case $0< \tfrac{{\varepsilon _1 }}{{\varepsilon _2 }}< \tfrac{3}{2}$ either. Our method of proof consists essentially of constructing a Dulac function (see 6] and 9]) and using the conception of Duff's rotated vector field (see 4], 7], 8], 10], and 11]). |
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