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. 1975 204 113 135 P. Habets and L. Sanchez, Periodic solutions of dissipative dynamical systems with singular potentials

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References [1] Aizicovici , S. , McKibben , M. , Reich , S. 2001 Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities Nonlinear Anal

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-004-4292-2 . [28] Zhou , Z. , Yu , J. S. , Chen , Y. M. 2010 Periodic solutions of a 2 n th-order nonlinear difference equation Sci. China Math. 53 41 – 50 10.1007/s11425-009-0167-7 .

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. , Y u , J. S. and G uo , Z. M. , Existence of periodic solutions for fourthorder difference equations , Comput. Math. Appl. , 50 ( 1–2 ) ( 2005 ), 49 – 55 . [8] C

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Abstract  

The aim of this paper is to investigate sufficient conditions (Theorem 1) for the nonexistence of nontrivial periodic solutions of equation (1.1) withp ≡ 0 and (Theorem 2) for the existence of periodic solutions of equation (1.1).

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Abstract  

In the present paper the method of the equivalent differential-operator equation has been applied in the study of the existence and asymptotic representation of periodic solutions of autonomous systems of the form

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Abstract  

We discuss two techniques useful in the investigation of periodic solutions of broad classes of non-linear non-autonomous ordinary differential equations, namely the trigonometric collocation and the method based upon periodic successive approximations.

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The paper applies a numerical-analytical method for finding periodic solutions of the system of integro-differential equations
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\begin{gathered} \dot x = f(t,x,\mathop \smallint \limits_0^t \varphi (t,s,x(s))ds), t \ne t_i (x), \hfill \\ \Delta x|_{t = t_i (x)} = I_i (x). \hfill \\ \end{gathered}$$ \end{document}
Two theorems for existence of periodic solutions are proved for the cases whent = t i andt = t i(x).
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In [1] (p. 215), the authors Andronov, Leontovich-Andronova, Gordon, and Maier, consider the following equation:
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\{ \begin{gathered} \tfrac{{dx}}{{dt}} = y, \hfill \\ \tfrac{{dy}}{{dt}} = x + x^2 - \left( {\varepsilon _1 + \varepsilon _2 x} \right)y, \hfill \\ \end{gathered} \right.$$ \end{document}
wheree 1 ande 2 are real constants ande 1 ande 2 are not both zero. They proved that there are no non-trivial periodic solutions except possibly for the case . 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 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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