Gevrey solutions of singularly perturbed differential equations.

*(English)*Zbl 0937.34075Consider a system of \(n\) singularly perturbed differential equations
\[
\varepsilon^\sigma \frac{dy}{dx}= F(x,y,a,\varepsilon)
\]
where \(x\) is a complex variable, \(\varepsilon\) a small complex parameter, \(\sigma\) a positive integer, \(a\) a vector of additional parameters and \(F\) is an analytic function. The authors study the existence of well behaved so-called overstable “solutions” to this system. Such an overstable solution is a family of solutions to a family of systems, more precisely a couple \((y^* (x,\varepsilon), a^* (\varepsilon))\), where \(a^* (\varepsilon)\to a_0\) as \(\varepsilon\to 0\) and the (family of) function(s) \(y^* (x, \varepsilon)\) satisfies \(\varepsilon^\sigma\frac{dy^*}{dx}= F(x, y^*, a^*, \varepsilon)\) and remains bounded as \(\varepsilon\to 0\) uniformly in \(x\) in a full neighborhood of some point \(x_0\). More precisely, they study overstable solutions that tends, as \(\varepsilon\to 0\), to some given \(\varphi_0(x)\). Here, \(\varphi_0 (x)\) is a so-called slow curve of the system, it has to satisfy \(F(x, \varphi_0(x), a_0, 0)= 0\).

Questions of this nature appear in the study of phenomena like bifurcation delay, canard solutions, adiabatic invariants and the existence of fundamental solutions with given asymptotic expansions (this is connected to an important conjecture of W. Wasow). The authors suppose that the Jacobian \(A_0(x)= \frac{\partial F}{\partial y}(x, \varphi_0(x), a_0, 0)\) is not identically zero, but they allow a zero of order \(m\) at \(x_0\). Beginning with a particular case, they consider only problems with parameters \(a\in \mathbb{C}^m\). If \(m>0\), they suppose that the dependence of \(F\) upon \(a\) satisfies a certain (natural) transversality condition. As a first step, it is proved the existence of formal solutions (in \(\varepsilon\)) \((\widehat{y} (x, \varepsilon), \widehat{a} (\varepsilon))\) and established their Gevrey character. Then using this result, a formal Borel transform and a truncated Laplace transform, they construct “quasi-solutions” (which, roughly speaking, solve the equation except for an exponentially small correction term). The existence of actual overstable solutions on full neighborhoods of \(x_0\) follows. It is also shown that two overstable solutions only differ by exponentially small terms. The authors end with their most general results about the existence of overstable solutions in the general case \((a\in \mathbb{C}^s\)) and prove some conjectures of G. Wallet. These results generalize some previous results due (separately) to the same authors and G. Wallet.

Questions of this nature appear in the study of phenomena like bifurcation delay, canard solutions, adiabatic invariants and the existence of fundamental solutions with given asymptotic expansions (this is connected to an important conjecture of W. Wasow). The authors suppose that the Jacobian \(A_0(x)= \frac{\partial F}{\partial y}(x, \varphi_0(x), a_0, 0)\) is not identically zero, but they allow a zero of order \(m\) at \(x_0\). Beginning with a particular case, they consider only problems with parameters \(a\in \mathbb{C}^m\). If \(m>0\), they suppose that the dependence of \(F\) upon \(a\) satisfies a certain (natural) transversality condition. As a first step, it is proved the existence of formal solutions (in \(\varepsilon\)) \((\widehat{y} (x, \varepsilon), \widehat{a} (\varepsilon))\) and established their Gevrey character. Then using this result, a formal Borel transform and a truncated Laplace transform, they construct “quasi-solutions” (which, roughly speaking, solve the equation except for an exponentially small correction term). The existence of actual overstable solutions on full neighborhoods of \(x_0\) follows. It is also shown that two overstable solutions only differ by exponentially small terms. The authors end with their most general results about the existence of overstable solutions in the general case \((a\in \mathbb{C}^s\)) and prove some conjectures of G. Wallet. These results generalize some previous results due (separately) to the same authors and G. Wallet.

Reviewer: J.P.Ramis

##### MSC:

34M60 | Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent) |

34M25 | Formal solutions and transform techniques for ordinary differential equations in the complex domain |