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We consider a singularly perturbed complex first order ODE eu ' = Φ(x, u, a, e), x, u , e > 0 is a small complex parameter and a is a control parameter. It is proven that the singularities of some solutions are regularly spaced and that they move from one to the next as a runs about a loop of index one around a value of overstability. This gives a positive answer to a question of J. L. Callot.

We consider the second-order differential equation e2y″ e (1+e2ψ(x, e))y with a small parameter e, where ψ is even with respect to e. It is well known that it has two formal solutions y±(x, e) e e±x/eh±(x, e), where h±(x, e) is a formal series in powers of e whose coefficients are functions of x. It has been shown l4r that one resp. both of these solutions are 1-summable in certain directions if ψ satisfies certain conditions, in particular, concerning its x-domain. In the present article we give necessary (and sufficient) conditions for 1-summability of one or both of the above formal solutions in terms of ψ. The method of proof involves a certain inverse problem, i.e., the construction of a differential equation of the above form exhibiting a prescribed Stokes phenomenon with respect to e.

Consider the systems (1) \ (sI-B){dv \over ds} = (\rho I-A)v and (2) \ {dx \over dt} = (B+t^{-1} A)x, where s and t are variables, ρ is a parameter, and A and B e diag(λ1,…,λn) are n by n matrices. (1) has only regular singular points, and (2) has an irregular singular point at t e ∞. Several kinds of special solutions having particular behavior near singular points were selected in previous papers. In the present paper, the author shows how (2) results from (1) in a process of confluence as ρ → ∞. It is analyzed how the special solutions of (1) converge to those of (2) in that process. As a consequence new proofs of earlier results about connection problems are obtained.

We consider systems of linear differential equations near an irregular singularity of rank one with a leading coefficient matrix having all distinct eigenvalues. The main results express connection coefficients (Stokes' multipliers and connection factors) as sums of convergent series or as limits of sequences, where the terms may be calculated in various ways. One way is to form certain explicit weighted sums of the coefficients in the formal or Frobenius series solutions. Another way is to recursively solve for the coefficients of a solution of a certain singular integral equation similar to the Frobenius method for differential equations near a regular singular point. In case the original differential equation has rational coefficients, then associated linear differential equations with rational coefficients can be constructed whose Frobenius solution determines die terms in the series or limits; hence all transformations are done within the class of “holonomic” functions.

Solutions of the vector difference equationy(x+e)−y(x−e)=2ef(x,y(x)),x being a complex variable and e>0 a small parameter, are constructed that are analytic onx-domains Ω which are independent of e. As the first case, horizontally convex bounded domains are considered, i.e., domains having the property that for eachx, x′∈Ω with the same imaginary part, the interval [x, x′] is contained in Ω; also considered are unbounded domains such as sectors open on the left or on the right. Using these results, it is shown that the Hausdorff distance between separatrices of certain systems of difference equations is exponentially small with respect to e. As an application, the so-called ghost solutions of the discretized logistic equation are considered in detail and, in particular, the lengths of the levels are estimated. Other applications, e.g., to the standard mapping, are presented.

In this note we will be concerned with nonhomogeneous linear difference equations of the form where a(z) and b(z) are assumed to be analytic in a neighborhood of infinity and vanish there, but we consider affine transformations , where r(z) and s(z) have convergent factorial series expansions in a certain left or right half-plane, together with translations of the independent variable and we ask for (in some sense) the simplest difference equations that a given one of the above from can be transformed into using these operations, i.e. we ask for canonical representatives from each distinct equivalence class. We will show that our difference equation is reucible to a difference equation with only 1 parameter in right half-planes and to one with only 2 parameters in left half-planes; one of these parametres is a1.

First, in joint work with S. Bodine of the University of Puget Sound, Tacoma, Washington, USA, we consider the second-order differential equation e2 y''=(1+e2 ψ(x, e))y with a small parameter e, where ψ is analytic and even with respect to e. It is well known that it has two formal solutions of the form y±(x,e)=e±x/eh±(x,e), where h±(x,e) is a formal series in powers of e whose coefficients are functions of x. It has been shown that one (resp. both) of these solutions are 1-summable in certain directions if ψ satisfies certain conditions, in particular concerning its x-domain. We show that these conditions are essentially necessary for 1-summability of one (resp. both) of the above formal solutions. In the proof, we solve a certain inverse problem: constructing a differential equation corresponding to a certain Stokes phenomenon. The second part of the paper presents joint work with Augustin Fruchard of the University of La Rochelle, France, concerning inverse problems for the general (analytic) linear equations e r y' = A(x,e) y in the neighborhood of a nonturning point and for second-order (analytic) equations e y'' - 2xy'-g(x,e) y=0 exhibiting resonance in the sense of Ackerberg-O'Malley, i.e., satisfying the Matkowsky condition: there exists a nontrivial formal solution $$\hat y\left( {x{\text{, }}\varepsilon } \right) = \sum {y_n } \left( x \right)\varepsilon ^n $$ such that the coefficients have no poles at x=0.

In recent years, the theory of Borel summability or multisummability of divergent power series of one variable has been established and it has been proved that every formal solution of an ordinary differential equation with irregular singular point is multisummable. For partial differential equations the summability problem for divergent solutions has not been studied so well, and in this paper we shall try to develop the Borel summability of divergent solutions of the Cauchy problem of the complex heat equation, since the heat equation is a typical and an important equation where we meet diveregent solutions. In conclusion, the Borel summability of a formal solution is characterized by an analytic continuation property together with its growth condition of Cauchy data to infinity along a stripe domain, and the Borel sum is nothing but the solution given by the integral expression by the heat kernel. We also give new ways to get the heat kernel from the Borel sum by taking a special Cauchy data.

Given a system of linear differential equations near an irregular singularity of pole type, formal invariants are quantities that remain unchanged with respect to linear transformations of the system. While certain “natural” formal invariants can easily be observed in formal fundamental solution matrices, the algorithms for constructing them do not readily show how the invariants can be universally described as properties of the coefficient matrix of the system, and in particular of the individual constant matrices in the power-series expansion. Other invariants have been abstractly defined by mapping properties of the differential operator, but they are not immediately related to either the natural invariants or the coefficients. In this paper we show how certain invariants in the formal solution may be described and calculated through matrix-theoretic properties of the coefficients and at the same time show how they are related to ones for the differential operator.

SynopsisWe give a direct and simple proof of a central result of the above paper.

The purpose of this dissertation is to study how the discretization of a differential equation affects, the stable and unstable manifolds in two concrete examples: the logistic equation and the pendulum equation. The logistic equation is equivalent to a system with two fixed points A and B. It is known that the stable manifolds at A coincides with the unstable manifold at B. By improving some results of A. Fruchard and R. Schäfke, we show that the two manifolds do not coincide any more in the discretezed equation. The proof is a modification of an approach introduced by R. Schäfke and H. Volkmer. First, we build a formal solution with polynomial coefficients. Then we give an asymptotic approximation of these coefficients. From these estimates we can obtain a quasi-solution, that is, a function which satisfies the difference equation except for an exponentially small error; moreover we can evaluate the asymptotic behavior of the distance between the two manifolds. To conclude, we show that some constant alpha appearing in the dominant term of the distance between the manifolds is non zero, and we further give a precise approximation for it. The second part of the thesis is dedicated to an analogous study regarding the pendulum equation and its discretization (Standard mapping). Similar results were obtained by Lazutkin et al., but our proof is completely different. This case is harder than the previous one, for we deal with a second order equation., Cette thèse a pour objet d'étudier l'influence de la discrétisation d'une équation différentielle sur les variétés stables et instables dans deux exemples concrets : l'équation logistique et l'équation du pendule. L'équation logistique est équivalente à un système qui admet deux points selles A et B. Il est connu que la variété stable en A coïncide avec la variété instable en B. En améliorant des résultats antérieures de A. Fruchard et R. Schäfke, nous montrons que les deux variétés ne coïncident plus pour l'équation discrétisée. La démonstration est basée sur une modification d'une approche développée par R. Schäfke et H. Volkmer. Nous construisons d'abord une solution formelle à coefficients polynomiaux. Ensuite, nous donnons une approximation asymptotique des coefficients de la solution formelle. Ces estimations nous permettent d'obtenir une quasi-solution c'est à dire une fonction qui vérifie l'équation aux différences avec une erreur exponentiellement petite, puis de déterminer le comportement asymptotique de la distance entre les deux variétés. Pour conclure, nous démontrons qu'une constante alpha dans le terme dominant de la distance entre les variétés n'est pas nulle et nous donnons une approximation précise de cette constante. La deuxième partie de cette thèse est consacrée à une étude analogue concernant l'équation du pendule et de sa discrétisation (Application standard). Des résultats similaires ont été obtenus par Lazutkin et al., mais la preuve que nous avons utilisée est complètement différente. Ce cas est plus difficile que le précédent parce qu'il s'agit d'une équation du second ordre.

The purpose of this dissertation is to study how the discretization of a differential equation affects, the stable and unstable manifolds in two concrete examples: the logistic equation and the pendulum equation. The logistic equation is equivalent to a system with two fixed points A and B. It is known that the stable manifolds at A coincides with the unstable manifold at B. By improving some results of A. Fruchard and R. Schäfke, we show that the two manifolds do not coincide any more in the discretezed equation. The proof is a modification of an approach introduced by R. Schäfke and H. Volkmer. First, we build a formal solution with polynomial coefficients. Then we give an asymptotic approximation of these coefficients. From these estimates we can obtain a quasi-solution, that is, a function which satisfies the difference equation except for an exponentially small error; moreover we can evaluate the asymptotic behavior of the distance between the two manifolds. To conclude, we show that some constant alpha appearing in the dominant term of the distance between the manifolds is non zero, and we further give a precise approximation for it. The second part of the thesis is dedicated to an analogous study regarding the pendulum equation and its discretization (Standard mapping). Similar results were obtained by Lazutkin et al., but our proof is completely different. This case is harder than the previous one, for we deal with a second order equation.; Cette thèse a pour objet d'étudier l'influence de la discrétisation d'une équation différentielle sur les variétés stables et instables dans deux exemples concrets : l'équation logistique et l'équation du pendule. L'équation logistique est équivalente à un système qui admet deux points selles A et B. Il est connu que la variété stable en A coïncide avec la variété instable en B. En améliorant des résultats antérieures de A. Fruchard et R. Schäfke, nous montrons que les deux variétés ne coïncident plus pour l'équation discrétisée. La démonstration est basée sur une modification d'une approche développée par R. Schäfke et H. Volkmer. Nous construisons d'abord une solution formelle à coefficients polynomiaux. Ensuite, nous donnons une approximation asymptotique des coefficients de la solution formelle. Ces estimations nous permettent d'obtenir une quasi-solution c'est à dire une fonction qui vérifie l'équation aux différences avec une erreur exponentiellement petite, puis de déterminer le comportement asymptotique de la distance entre les deux variétés. Pour conclure, nous démontrons qu'une constante alpha dans le terme dominant de la distance entre les variétés n'est pas nulle et nous donnons une approximation précise de cette constante. La deuxième partie de cette thèse est consacrée à une étude analogue concernant l'équation du pendule et de sa discrétisation (Application standard). Des résultats similaires ont été obtenus par Lazutkin et al., mais la preuve que nous avons utilisée est complètement différente. Ce cas est plus difficile que le précédent parce qu'il s'agit d'une équation du second ordre.