×

Exponentially small splitting of separatrices for difference equations with small step size. (English) Zbl 0940.39001

Summary: Solutions of the vector difference equation \(y(x+ \varepsilon)-y(x-\varepsilon)=2 \varepsilon f(x,y(x))\), \(x\) being a complex variable and \(\varepsilon>0\) a small parameter, are constructed that are analytic on \(x\)-domains \(\Omega\) which are independent of \(\varepsilon\). As the first case, horizontally convex bounded domains are considered, i.e., domains having the property that for each \(x,x'\in\Omega\) with the same imaginary part, the interval \([x,x']\) is contained in \(\Omega\); 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 \(\varepsilon\). 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.

MSC:

39A10 Additive difference equations
39A12 Discrete version of topics in analysis
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. Fruchard, La somme d’une fonction.Preprint IRMA 7, rue René Descartes 67084Strasbourg Cedex, France, 1993.
[2] E. Noerlund, Sur la ”Somme” d’une fonction.Mém. Sc. Math. 24 (1927).
[3] –, Vorlesungen über Differenzenrechnung.Chelsea, New York, 1954.
[4] W.A. Harris and Y. Sibuya, On asymptotic solutions of systems of nonlinear difference equations.J. Reine Angew. Math. 222 (1966), 120–135. · Zbl 0142.05703 · doi:10.1515/crll.1966.222.120
[5] G. Immink, Asymptotics of analytic difference equations.Lect. Notes Math. 1085 (1984);Springer-Verlag, 1981, 618–626.
[6] E. Fontich and C. Simó, The splitting of separatrices for analytic diffeomorphisms.Ergod. Theory and Dynam. Syst. 10 (1990), 295–318. · Zbl 0706.58061
[7] M. Yamaguti and S. Ushiki, Chaos in numerical analysis of ordinary differential equations.Physica D 3 (1981), 618–626. · Zbl 1194.37064 · doi:10.1016/0167-2789(81)90044-0
[8] S.N. Chow, E.M. de Jager, and R. Lutz, The ghost solutions of the logistic equation and a singular perturbation problem. In: Advances in Computational Methods for Boundary and Interior Layers.Dublin, 1984, 15–20.
[9] T. Sari, Stroboscopy, averaging and long time behaviour in dynamical systems.Preprint Université de Sidi Bel Abbès, 190,Algérie.
[10] V.F. Lazutkin, I.G. Schachmannski, and M.B. Tabanov, Splitting of separatrices for standard and semistandard mappings.Physica D 40 (1989), 235–248. · Zbl 0825.58033 · doi:10.1016/0167-2789(89)90065-1
[11] A. Fruchard, Les fonctions périodiques de période infiniment petite.C. R. Acad. Sci. Ser. 1318 (1994), 227–230. · Zbl 0799.39006
[12] E. Fontich and C. Simo, Invariant manifolds for near identity maps and splitting of separatrices.Ergod. Theory and Dynam. Syst. 10 (1990), 319–346.
[13] E.A. Coddington and N. Levinson, Theory of ordinary differential equations.McGraw-Hill, New York, 1955. · Zbl 0064.33002
[14] B.V. Chirikov, A universal instability of many dimensional oscillator systems.Phys. Rep. 52 (1979), 263–279. · doi:10.1016/0370-1573(79)90023-1
[15] V. Hakim and K. Mallick, Exponentially small splitting of separatrices, matching in the complex plane and Borel summation.Nonlinearity 6 (1993), 57–70. · Zbl 0769.34036 · doi:10.1088/0951-7715/6/1/004
[16] E. Hille, Analytic function theory. Vol. 2.Ginn, Boston, 1962. · Zbl 0102.29401
[17] W. Eckhaus, Asymptotic analysis of singular perturbations.North-Holland, Amsterdam, 1979. · Zbl 0421.34057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.