Nonlinear Stokes phenomena in first or second order differential equations

We study singularity formation in nonlinear differential equations of order m ≤ 2, y (m) = A(x −1, y). We assume A is analytic at (0, 0) and ∂ y A(0, 0) = λ ≠ = 0 (say, λ = (−1) m ). If m = 1 we assume A(0, ·) is meromorphic and nonlinear. If m = 2, we assume A(0, ·) is analytic except for isolated singularities, and also that ∫ s0 ∞ |Ф(s)|−1/2 d|s| a > 0, arg(z) ∈ (−α, α)}. If the Stokes constant S + associated to ℝ+ is nonzero, we show that all y such that lim x→+∞ y(x) = 0 are singular at 2πi-quasiperiodic arrays of points near iℝ+. The array location determines and is determined by S +. Such settings include the Painlevé equations P I and P II . If S + = 0, then there is exactly one solution y 0 without singularities in H2π−∈, and y 0 is entire iff y 0 = A(z, 0) ≡ 0. The singularities of y(x) mirror the singularities of the Borel transform of its asymptotic expansion, $$ \mathcal{B}\tilde y $$ , a nonlinear analog of Stokes phenomena. If m = 1 and A is a nonlinear polynomial with A(z, 0) ≢ 0 a similar conclusion holds even if A(0, ·) is linear. This follows from the property that if f is superexponentially small along ℝ+ and analytic in H π, then f is superexponentially unbounded in H π, a consequence of decay estimates of Laplace transforms. Compared to [2] this analysis is restricted to first and second order equations but shows that singularities always occur, and their type is calculated in the polynomial case. Connection to integrability and the Painlevé property are discussed.