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Overstability and resonance

Augustin Fruchard;  Reinhard Schäfke — 2003

Annales de l’institut Fourier

We consider a singularity perturbed nonlinear differential equation ε u ' = f ( x ) u + + ε P ( x , u , ε ) which we suppose real analytic for x near some interval [ a , b ] and small | u | , | ε | . We furthermore suppose that 0 is a turning point, namely that x f ( x ) is positive if x ≠ 0 . We prove that the existence of nicely behaved (as ϵ → 0 ) local (at x = 0 ) or global, real analytic or C ∞ solutions is equivalent to the existence of a formal series solution ∑ u n ( x ) ε n with u n analytic at x = 0 . The main tool of a proof is a new “principle of analytic continuation” for such “overstable” solutions....

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