
handle: 1942/22725
In this article Gevrey properties of analytic invariant curves of analytic slow-fast systems \[ \begin{aligned} \dot{x} & =\varepsilon x, \\ \dot{y} & = \varphi(x)y+\varepsilon H(x,y,\varepsilon) \end{aligned} \] with \(\varphi(0)<0\) are studied. The formal invariant manifold is locally given as a series \(y=\sum_{n=0}^\infty y_n(x)\varepsilon^n\) that is Gevrey-1 in \(\varepsilon\). Using that the Borel transform of this formal solution coincides in some sector with an exact solution of at most exponential growth suffices to prove the 1-summability with respect to \(\varepsilon\) in the positive real direction. Based on this result the author then studies canard solutions that are generated if the two invariant manifolds from two different slow-fast saddle points are connected near a turning point of the slow manifold. The main theorem states that for a real analytic slow-fast system a heteroclinic saddle connection between two slow-fast saddle equilibria on the slow manifold is summable with respect to the singular parameter in the positive real direction uniformly for \(x\) in compact subsets of the domain of the saddle connection which do not include the turning point.
formal solution, Gevrey series, Gevrey series; Borel summation; Slow-fast systems; Singular perturbations; Canards, Borel transform, Canard solutions to ordinary differential equations, Gevrey series; Borel summation; slow-fast systems; singular perturbations; canards, Perturbations, asymptotics of solutions to ordinary differential equations, slow-fast systems, Borel summation, Singular perturbations for ordinary differential equations, canards, singular perturbations, Asymptotic expansions of solutions to ordinary differential equations, Invariant manifolds for ordinary differential equations
formal solution, Gevrey series, Gevrey series; Borel summation; Slow-fast systems; Singular perturbations; Canards, Borel transform, Canard solutions to ordinary differential equations, Gevrey series; Borel summation; slow-fast systems; singular perturbations; canards, Perturbations, asymptotics of solutions to ordinary differential equations, slow-fast systems, Borel summation, Singular perturbations for ordinary differential equations, canards, singular perturbations, Asymptotic expansions of solutions to ordinary differential equations, Invariant manifolds for ordinary differential equations
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