On the application of geometric singular perturbation theory to some classical two point boundary value problems

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 8, Issue 2, 1998, Pages 189-209

  Hayes, Michael ^a   Kaper, Tasso J ^a   Kopell, Nancy ^a   Ono, Kinya ^a  

^a Boston University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032008350     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127498000140     Document Type: Article

References (47)

1. Angenent, S. B., Mallet-Paret, J. & Peletier, L. A. [1987] "Stable transition layers in semilinear boundary value problems," J. Diff. Eq. 67, 212-242.
2. Bender, C. M. & Orszag, S. A. [1978] Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York).
3. Camassa, R., Kovacic, G. & Tin, S. K. [1996] "A Melnikov method for homoclinic orbits with many pulses," (preprint).
4. Chang, K. W. & Howes, F. A. [1984] Nonlinear Singular Perturbation Phenomena: Theory and Applications (Springer-Verlag, New York).
5. Diener, F. & Diener, M. [1996] Nonstandard Analysis in Practice (Springer-Verlag, New York).
6. Eckhaus, W. [1973] Matched Asymptotic Expansions and Singular Perturbations (North-Holland, Amsterdam).
7. Eckhaus, W. [1979] Asymptotic Analysis of Singular Perturbations (North-Holland, Amsterdam).
8. Eckhaus, W. [1994] "Fundamental concepts of matching," SIAM Rev. 36, 431-439.
9. Fenichel, N. [1978] "Center manifolds in bifurcation theory and singular perturbation theory," Institute of Applied Mathematics and Statistics, University of British Columbia, Technical Report 78(31).
10. Fenichel, N. [1979] "Geometric singular perturbation theory for ordinary differential equations," J. Diff. Eq. 31, 53-98.
11. Fife, P. C. [1974] "Transition layers in singular perturbation problems," J. Diff. Eq. 15, 77-105.
12. Fife, P. C. [1979] Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28 (Springer-Verlag, New York).
13. Grasman, J. [1987] Asymptotic Methods of Relaxation Oscillations and Applications (Springer-Verlag New York).
14. Hirsch, M. W., Pugh, C. C. & Shub, M. [1977] Invariant Manifolds (Springer-Verlag, New York).
15. Holmes, M. H. [1995] Introduction to Perturbation Methods (Springer-Verlag, New York).
16. Hoppenstaedt, F. [1970] "Properties of solutions of ordinary differential equations with a small parameter," Comm. Pure Appl. Math. 24, 807-840.
17. Howes, F. A. [1979] Boundary-Interior Layer Interactions in Nonlinear Singular Perturbation Theory, Memoirs AMS. 203 (American Mathematical Society, Providence).
18. Jones, C. K. R. T., Kopell, N. & Langer, R. [1991] "Construction of the FitzHugh-Nagumo pulse using differential forms," in Patterns and Dynamics in Reactive Media, eds. Swinney, H., Aris, G. & Aronson, D., IMA volumes in Mathematics and its Applications 37 (Springer-Verlag, New York), pp. 101-116.
19. Jones, C. K. R. T. [1994] "Geometric singular perturbation theory," in Dynamical Systems, Montecatini Terme, ed. Arnold, L., Lecture Notes in Mathematics 1609 (Springer-Verlag, Berlin).
20. Jones, C. K. R. T. & Kopell, N. [1994] "Tracking invariant manifolds with differential forms," J. Diff. Eq. 108, 64-88.
21. Jones, C. K. R. T., Kaper, T. J. & Kopell, N. [1996] "Tracking invariant manifolds up to exponentially small errors," SIAM J. Math. Anal. 27, 558-577.
22. Kaper, T. J. & Kovacic, G. [1996] "Multibump orbits homoclinic to resonance bands," Trans. AMS. 348, 3835-3887.
23. Kath, W. L. [1985] "Slowly varying phase planes and boundary-layer theory," Stud. Appl. Math 72, 221-239.
24. Keller, H. B. [1976] Numerical Solution of Two Point Boundary Value Problems, Regional Conference Series in Applied Mathematics (SIAM, Philadelphia).
25. Kevorkian, J. & Cole, J. D. [1981] Perturbation Methods in Applied Mathematics (Springer-Verlag, New York).
26. Kokubu, H., Mischaikow, K. & Oka, H. [1996] "Existence of infinitely many connecting orbits in a singularly perturbed ordinary differential equation," Nonlinearity 9, 1263-1280.
27. Kopell, N. [1980] "The singularly perturbed turning-point problem: A geometric approach," in Singular Perturbations and Asymptotics, eds. Meyer, R. E. & Porter, S. V. (Academic Press, New York).
28. Kopell, N. & Landman, M. [1995] "Spatial structure of the focusing singularity of the nonlinear Schrödinger equation: A geometrical analysis," SIAM J. Appl. Math. 55(5), 1297-1323.
29. Lagerstrom, P. A. [1988] Matched Asymptotic Expansions: Ideas and Techniques (Springer-Verlag New York).
30. Lin, X. B. [1990] "Heteroclinic bifurcation and singularly perturbed boundary value problems," J. Diff. Eq. 84, 319-382.
31. Mischenko, E. F. & Rozov, N. Kh. [1980] Differential Equations with Small Parameters and Relaxation Oscillations (Plenum Press, New York).
32. Murdock, J. A. [1991] Perturbations: Theory and Methods (Wiley, New York).
33. Nayfeh, A. H. [1973] Perturbation Methods (Wiley New York).
34. Nipp, K. [1985] "Invariant manifolds of singularly perturbed ordinary differential equations," Zeitschrift für Angewandte Mathematik und Physik 36, 309-320.
35. O'Malley, R. E., Jr. [1974] Introduction to Singular Perturbations (Academic Press, New York).
36. O'Malley, R. E., Jr. [1991] Singular Perturbation Methods for Ordinary Differential Equations (Springer-Verlag, New York).
37. O'Malley, R. E., Jr. [1994] "On singular perturbations, especially matching," SIAM Rev. 36(3), 413-414.
38. Sakamoto, K. [1990] "Invariant manifolds in singular perturbation problems for ordinary differential equations," Proc. Royal Soc. Edin. 116A 45-78.
39. Smith, D. R. [1985] Singular Perturbation Theory: An Introduction with Applications (Cambridge University Press, New York).
40. Soto-Treviño, C. & Kaper, T. J. [1996] "Higher-order Melnikov theory for adiabatic systems," J. Math. Phys. 37, 6220-6249.
41. Takens, F. [1971] "Partially hyperbolic fixed points," Topology 10, 133-147.
42. Tikhonov, A. N. [1948] "On the dependence of the solutions of differential equations on a small parameter," Mat. Sb. 31, 575-586.
43. Tin, S. K. [1994] "On the dynamics of tangent spaces near a normally hyperbolic invariant manifold," PhD thesis, Brown University.
44. Tin, S. K., Kopell, N. & Jones, C. K. R. T. [1994] "Invariant manifolds and singularly perturbed boundary value problems," SIAM J. Num. Anal. 31, 1558-1576.
45. Van Dyke, M. [1964] Perturbation Methods in Fluid Mechanics (Academic Press, New York).
46. Wasow, W. R. [1956] "Singular perturbations of boundary value problems for nonlinear differential equations of second-order," Comm. Pure Appl. Math. 9, 93-116.
47. Wiggins, S. [1994] Normally Hyperbolic Invariant Manifolds in Dynamical Systems (Springer-Verlag, New York).