An algorithm to obtain global solutions of the double confluent Heun equation

A procedure is proposed to construct solutions of the double confluent Heun equation with a determinate behaviour at the singular points. The connection factors are expressed as quotients of Wronskians of the involved solutions. Asymptotic expansions are used in the computation of those Wronskians. The feasibility of the method is shown in an example, namely, the Schroedinger equation with a quasi-exactly-solvable potential

Bifurcation delay and difference equations

International audienceWe prove the existence of complex analytic solutions of difference equations of the form $y(x+eps)=f(x,y(x))$, where x and y are complex variables and $eps$ is a small parameter.We also show that differences of two solutions are exponentially small.We apply these results to the problem of delayed bifurcation at a point of period doubling for real discrete dynamical systems. In contrast to previous publications,the results obtained in this article are global

Classification of resonant equations

International audienceWe consider singularly perturbed linear ordinary differential equations of the second order with coefficients analytic near some point, say 0. We assume that the coefficients are real valued on the real axis, i.e. that there is a turning point at the origin. Such equations are called resonant in the sense of Ackerberg-O'Malley, if there is a solution, analytic in some neighborhood of 0, which tends to a non-zero limit as the parameter tends to 0. The article presents a classification of such resonant equations with respect to linear transformations having analytic coefficients. Besides a formal invariant (considered fixed below), we associate three formal series of Gevrey order 1 to any resonant equation which are invariant under analytic transformations. It is shown that this correspondence between equivalence classes of resonant equations and triples of Gevrey 1 series is essentially bijective, and that each equivalence class contains an equation of a particular form

Exponentially Small Splitting of Separatrices for Difference Equations With Small Step Size

Solutions of the vector difference equation y(x + ") \Gamma y(x \Gamma ") = 2"f (x; y(x)), x a complex variable, " ? 0 a small parameter, are constructed that are analytic on x--domains\Omega independent of ". As a first case, horizontally convex bounded domains are considered, i.e. domains having the property that for each x; x 0 2\Omega with same imaginary part, the segment [x; x 0 ] is contained in \Omega\Gamma also considered are unbounded domains such as sectors open to the left or 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 ". 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. Abbreviated title: Exponentially small splitting of separatrices 1 Introduction Consid..