A Primer on the Exchange Lemma for Fast-Slow Systems

  • Tasso J. Kaper
  • Christopher K. R. T. Jones
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 122)


In this primer, we give a brief overview of the Exchange Lemma for fast-slow systems of ordinary differential equations. This Lemma has proven to be a useful tool for establishing the existence of homoclinic and heteroclinic orbits, especially those with many components or jumps, in a variety of traveling wave problems and perturbed near-integrable Hamiltonian systems. It has also been applied to models in which periodic orbits and solutions of boundary value problems are sought, including singularly perturbed two-point boundary value problems. The Exchange Lemma applies to fast-slow systems that have normally hyperbolic invariant manifolds (that are usually center manifolds). It enables one to track the dynamics of invariant manifolds and their tangent planes while orbits on them are in the neighborhood of a normally hyperbolic invariant manifold. The end result is a closeness estimate in the C 1 topology of the tracked manifold to a certain submanifold of the normally hyperbolic’s local unstable manifold. We review the general version of the Exchange Lemma due to Tin [24] that treats problems in which there is both fast and slow evolution on the center manifolds. The main normal form used in the neighborhoods of the invariant manifolds is obtained from the persistence theory for normally hyperbolic invariant manifolds due to Fenichel [5, 6, 7]. The works of Jones and Tin [11, 15, 24] form the basis for this work, and [15, 24] contain full presentations of all of the results stated here.


Periodic Orbit Invariant Manifold Unstable Manifold Tangent Plane Homoclinic Orbit 
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  1. [1]
    A. Bose, Symmetric and antisymmetric pulses in parallel coupled nerve fibers, SIAM J. Appl. Math., 55 (1995), 1650–1674.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    R. Camass A, G. Kovacic, and S.-K. Tin, A Melnikov method for homoclinic orbits with many pulses, Arch. Rat. Mech. Anal., 143 (1998), 105–193.CrossRefGoogle Scholar
  3. [3]
    B. Deng, The Silnikov problem, exponential expansion, strong λ-Lemma, C 1 linearization, and homoclinic bifurcation, J. Diff. Eq., 79 (1989), 189–231.zbMATHCrossRefGoogle Scholar
  4. [4]
    A. Doelman, T.J. Kaper, and P. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity, 10 (1997), 523–563.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971), 193–226.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J., 23 (1974), 1109–1137; Asymptotic stability with rate conditions, II, Indiana Univ. Math. J., 26 (1977), 81–93.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    N. Fenichel, Geometrical singular perturbation theory for ordinary differential equations, J. Diff. Eq., 31 (1979), 53–98.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    G. Haller, Chaos near resonance, Applied Mathematical Sciences Series, 138 (1999) (Springer-Verlag, New York).zbMATHCrossRefGoogle Scholar
  9. [9]
    M. Hayes, T.J. Kaper, N. Kopell, and K. Ono, On the application of geometric singular perturbation theory to some classical two-point boundary value problems, Int. J. Bif. Chaos, 8 (1998), 189–209.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    M. Hirsch, C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, 583 (1977) (Springer-Verlag, New York).zbMATHGoogle Scholar
  11. [11]
    C.K.R.T. Jones, Geometric singular perturbation theory, in Dynamical systems, Montecatini Terme, Lecture Notes in Mathematics, R. Johnson, ed. (Springer-Verlag, New York), 1609 (1994), 44–118.Google Scholar
  12. [12]
    C.K.R.T. Jones, T.J. Kaper, and N. Kopell, Tracking invariant manifolds up to exponentially small errors, SIAM J. Math. Anal., 27 (1996), 558–577.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    C.K.R.T. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Diff. Eq., 108 (1994), 64–88.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    C.K.R.T. Jones, N. Kopell, and R. Langer, Construction of the FitzHugh-Nagumo pulse using differential forms, in Patterns and dynamics in reactive media, IMA volumes in Mathematics and its Applications, ]H. Swinney, R. Aris, and D. Aronson, eds. (Springer-Verlag, New York), 37 (1991), 101–116.CrossRefGoogle Scholar
  15. [15]
    C.K.R.T. Jones and S.K. Tin, Generalized Exchange Lemmas and orbits heteroclinic to invariant manifolds, submitted, Mem. AMS, 2000.Google Scholar
  16. [16]
    T.J. Kaper, An introduction to geometric methods and dynamical systems theory for singular perturbation problems, in Analyzing multiscale phenomena using singular perturbation methods, Proceedings of Symposia in Applied Mathematics, J. Cronin and R.E. O’Malley, Jr., eds. (American Mathematical Society, Providence, RI), 56 (1999), 85–132.Google Scholar
  17. [17]
    T.J. Kaper and G. Kovacic, Multi-bump orbits homoclinic to resonance bands, Trans. AMS, 348 (1996), 3835–3887.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    J. Palis and W. Demelo, Geometric theory of dynamical systems (Springer-Verlag, New York), 1982.zbMATHCrossRefGoogle Scholar
  19. [19]
    J.E. Rubin and C.K.R.T. Jones, Existence of standing pulse solutions to an inhomogeneous reaction-diffusion system, J. Dyn. Diff. Eq., 10 (1998), 1–35.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    K. Sakamoto, Invariant manifolds in singular perturbation problems for ordinary differential equations, Proc. Roy. Soc. Edin., A116 (1990), 45–78.MathSciNetCrossRefGoogle Scholar
  21. [21]
    C. Soto-Trevino and T.J. Kaper, Higher-order Melnikov theory for adiabatic systems, J. Math. Phys., 37 (1996), 6220–6249.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    C. Soto-Trevino, Geometric methods for periodic orbits in singularly perturbed systems, Ph.D. thesis, Boston University, 1998.Google Scholar
  23. [23]
    C. Soto-Trevino, A geometric method for periodic orbits in singularly perturbed systems, this volume, 2000.Google Scholar
  24. [24]
    S.K. Tin, On the dynamics of tangent spaces near a normally hyperbolic invariant manifold, Ph.D. thesis, Division of Applied Mathematics, Brown University, 1994.Google Scholar
  25. [25]
    S.K. Tin, C.K.R.T. Jones, and N. Kopell, Invariant manifolds and singularly perturbed boundary value problems, SI AM J. Num. Anal., 31 (1994), 1558–1576.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    S. Wiggins, Normally hyperbolic invariant manifolds in dynamical systems, Applied Mathematical Sciences Series (Springer-Verlag, New York), 105 (1994).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Tasso J. Kaper
    • 1
  • Christopher K. R. T. Jones
    • 2
  1. 1.Department of Mathematics & Center for BioDynamicsBoston UniversityBostonUSA
  2. 2.Division of Applied Mathematics & Lefschetz Center for Dynamical SystemsBrown UniversityProvidenceUSA

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