Abstract
Given any finite dimensional autonomous first order system of ordinary differential equations and any one step discretization method consistent of order p ≥ 1, the diffeomorphism associated to step size a is the period map of the autonomous system with a certain periodic forcing.The forcing amplitude is of order O(ε p) as ε tends to zero, and the forcing frequency is 1/ε. In this sense, the forcing is rapid. We give a proof of this lemma (from Fiedler and Scheurle [1991]. Also, based on recent work on splitting of separatrices for rapidly forced systems, we discuss consequences concerning the numerical simulation of the long-time behaviour of autonomous systems near homo-clinic orbits. Especially, we want to make the point that discretizing a homoclinic orbit, generically breaks the homoclinic orbit and can lead to chaos. This will be discussed for both Hamiltonian and general systems. Also, upper bounds in terms of ε for the splitting effects are presented. Under appropriate smoothness assumptions, those are of arbitrarily high algebraic order in ε, as ε tends to zero. In particular, in the analytic case, they are exponentially small. A proof of these results is outlined at the end of the paper.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Beyn W.J. [1987] The effect of discretizations of homoclinic orbits, in “Bifurcation: Analysis, Algorithms, Applications”, T. Kipper ei al. (eds.), Birkhäuser-Verlag, Basel, 1–8.
Beyn W.J. [1990] The numerical computation of connecting orbits, IMA J. Numer. Analysis 9, 379–405.
Delshams A. and Seara T.M. [1992] An asymptotic expansion for the splitting of separatrices of the rapidly forced pendulum, Comm. Math. Phys. 150, 433–463.
Ellison J.A., Kummer M. and Sâenz A.W. [1993] Transcendentally small transversality in the rapidly forced pendulum, J. Dyn. Diff. Equat. 5 (2), 241.
Fiedler B. and Scheurle J. [1991] Discretization of homoclinic orbits, rapid forcing and “invisible” chaos, to appear in Memoirs of the AMS.
Fontich E. [1993] Exponentially small upper bounds for the splitting of separatrices for high frequency periodic perturbation, J. Nonlinear Analysis 20 (6), 733–744.
Fontich E. and Simó C. [1990a] Splitting of separatrices for analytic diffeomorphisms, Ergod. Theor. Dyn. Sys. 10, 295–318.
Fontich E. and Simó C. [1990b] Invariant manifolds for near identity differentiable maps and splitting of separatrices, Ergod. Theor. Dyn. Sys. 10, 319–346.
Genecand Ch. [1993] Transversal homoclinic orbits near elliptic fixed points of area preserving diffeomorphisms of the plane, Dynamics Reported 2 (New Series), 1–30.
Holmes P.J., Marsden J.E. and Scheurle J. [1988] Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations, Cont. Math. 81, 213–244.
Hunter J. and Scheurle J. [1988] Existence of perturbed solitary wave solutions to a model equation for water waves, Physica D 32, 253–268.
Kopell N. [1968] PhD thesis, University of California, Berkeley.
Lazutkin V.F., Schachmannski I.G. and Tabanov M.B. [1989] Splitting of separatrices for standard and semistandard mappings, Physica D 40, 235–248.
Marsden J.E. [1992] Lectures on Mechanics, London Math. Soc. Lecture Notes Series 174, University Press, Cambridge.
Neisthadt A.I. [1984] The separation of motion in systems with rapidly rotating phase, J. Appl. Math. Mech. 48 (2), 133–139.
Palmer K.J. [1984] Exponential dichotomies and transversal homoclinic points, J. Diff. Equat. 55, 225–256.
Scheurle J. [1989] Chaos in a rapidly forced pendulum equation, Cont. Math. 97, 411–419.
Scheurle J., Marsden J.E. and Holmes P.J. [1991] Exponentially small estimates for separatrix splittings, in “Asymptotics beyond all orders”, H. Segur et al. (eds.), Plenum Press, New York, 187–195.
Zehnder E. [1973] Homoclinic points near elliptic fixed points, Comm. Pure Appl. Math. 26, 131–182.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Scheurle, J. (1995). Discretization of Autonomous Systems and Rapid Forcing. In: Dumas, H.S., Meyer, K.S., Schmidt, D.S. (eds) Hamiltonian Dynamical Systems. The IMA Volumes in Mathematics and its Applications, vol 63. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8448-9_22
Download citation
DOI: https://doi.org/10.1007/978-1-4613-8448-9_22
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-8450-2
Online ISBN: 978-1-4613-8448-9
eBook Packages: Springer Book Archive