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Study of superstable solutions of the van der Pol equation. (Études des solutions surstables de l’équation de van der Pol.) (French) Zbl 1077.34060

Summary: It is well known that the forced van der Pol equation \(\varepsilon\ddot u+ (u^2- 1)\dot u+ u= \alpha\) (where \(\dot u= {du\over dt}\) and \(\varepsilon\to 0\)) in a singularly perturbed differential equation having exceptional solutions (called canards) for some values of \(\alpha(\varepsilon)\). There are exactly two distinct complex canards solutions \((v^+,\alpha^+)\), \((v^-,\alpha^-)\) of the transformed equation \(\varepsilon v{dv\over du}= (1- u^2 )v+ \alpha- u\), which are bounded in wide open sets of \(\mathbb{C}\), including a whole half-plane containing the points \(u= 0\) and \(u= 1\). We first prove the existence of a uniform asymptotic development of the solution \(v^+\) in sectors centered at \(u=-1\), like \(\{0< \arg(x +1)< 2\pi/3\), and \(|x+ 1|> |X_1|\,|\varepsilon|^{1/3}\}\). Explicit asymptotic approximations for \(\alpha^+- \alpha^-\) as \(\varepsilon\to 0\) and, as \(n\to\infty\), of the \(a_n\) (of the asymptotic serie \(\widehat\alpha= \sum a_n\varepsilon^n\) corresponding to \(\alpha\)) can then be calculated.

MSC:

34E15 Singular perturbations for ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)
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References:

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