## 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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