Abstract
Homoclinic orbits in the fast dynamics of singular perturbation problems are usually analyzed by a combination of Fenichel's invariant manifold theory with general transversality arguments (see Ref. 29 and the Exchange Lemma in Ref. 16). In this paper an alternative direct approach is developed which uses a two-time scaling and a contraction argument in exponentially weighted spaces. Homoclinic orbits with one last transition are treated and it is shown how ε-expansions can be extracted rigorously from this approach. The result is applied to a singularity perturbed Bogdanov point in the FitzHugh–Nagumo system.
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REFERENCES
Arnold, V. I., Afrajmovich, V. S., Il'yashenko, Y. S., and Shil'nikov, L. P. (1994). Dynamical Systems V: Bifurcation Theory, Springer, New York.
Baer, S. M., and Erneux, T. (1986). Singular Hopf bifurcation to relaxation oscillations. SIAM J. Appl. Math. 46(5), 721–739.
Beyn, W.-J. (1990). The numerical computation of connecting orbits in dynamical systems. IMA J. Numer. Anal. 10, 379–405.
Beyn, W.-J. (1994). Numerical analysis of homoclinic orbits emanating from a Takens-Bogdanov point. IMA J. Numer. Anal. 14, 381–410.
Bohl, E. (1978). P-boundedness of inverses of nonlinear operators. ZAMM 58, 277–287.
Casten, R. G., Cohen, H., and Lagerstrom, P. A. (1975). Perturbation analysis of an approximation to the Hodgkin-Huxley theory. Q. Appl. Math. 32, 365–402.
Chay, T. R., and Rinzel, J. (1985). Bursting, beating, and chaos in an excitable membrane model. Biophys. J. 47, 357–366.
Chow, S.-N., Hale, J., and Mallet-Paret, J. (1980). An example of bifurcation to homoclinic orbits. J. Diff. Eq. 37, 351–373.
Chow, S.-N., and Lin, X.-B. (1990). Bifurcation of a homoclinic orbit with a saddle-node equilibrium. Diff. Integral Eq. 3, 435–466.
Diener, M. (1981). Étude générique des canards, IRMA, Strasbourg.
Dumortier, F., and Roussarie, R. (1996). Canard Cycles and Center Manifolds, AMS, Vol. 121,No. 577.
Eckhaus, W. (1983). Relaxation oscillations including a standard chase on french ducks. In Lecture Notes in Mathematics, Vol. 985, Springer, New York, pp. 449–494.
Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations. J. Diff. Eq. 31, 53–98.
Guckenheimer, J., and Holmes, P. (1990). Nonlinear Oscillations, Dynamical systems, and Bifurcations of Vector Fields (3rd printing), Appl. Math. Sci., Vol. 42, Springer, New York.
Hastings, S. P. (1982). Single and multiple pulse waves for the FitzHugh-Nagumo equations. SIAM J. Appl. Math. 42, 247–260.
Jones, C. K. R. T., and Kopell, N. (1994). Tracking invariant manifolds with differential forms in singularly perturbed systems. J. Diff. Eq. 108, 64–88.
Koper, M. (1995). Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram. Physica D 80, 72–94.
Krupa, M., Sandstede, B., and Szmolyan, P. (1997). Fast and slow waves in the FitzHugh-Nagumo equation. J. Diff. Eq. 133, 49–97.
Lin, X.-B. (1990). Using Melnikov's method to solve Silnikov's problems. Proc. Roy. Soc. Edinburgh 116A, 295–325.
Lin, X.-B. (1990). Heteroclinic bifurcation and singularly perturbed boundary value problems. J. Diff. Eq. 84, 319–382.
O'Malley, R. E. (1991). Singular Perturbation Methods for Ordinary Differential Equations, Applied Mathematical Sciences, Vol. 89, Springer, New York.
Palmer, K. J. (1984). Exponential dichotomies and transversal homoclinic points. J. Diff. Eq. 55, 225–256.
Sell, G. R. (1985). Smooth linearization near a fixed point. Am. J. Math. 107, 1035–1091.
Shub, M. (1987). Global Stability of Dynamical Systems, Springer, New York.
Stiefenhofer, M. (1995). Singuläre Störung und Verzweigung bei Schleimpilzen, Ph.D. thesis, University of Konstanz, Hartung-Gorre Verlag, Konstanz.
Stiefenhofer, M. (1996). Singular perturbation with limit points in the fast dynamics. DFG-Schwerpunktprogramm: Dynamik, Analysis, effiziente Simulation und Ergodentheorie. ZAMP 49, 730–758.
Stiefenhofer, M. (1996). Singular perturbation with Hopf points in the fast dynamics. DFG-Schwerpunktprogramm: Dynamik, Analysis, effiziente Simulation und Ergodentheorie. ZAMP 49, 602–649.
Stiefenhofer, M. (1997). Singular perturbation with Bogdanov points in the fast dynamics. DFG-Schwerpunktprogramm: Dynamik, Analysis, effiziente Simulation und Ergodentheorie. Preprint No. 10.
Szmolyan, P. (1991). Transversal heteroclinic and homoclinic orbits in singular perturbation problems. J. Diff. Eq. 92, 252–281.
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Supported by DFG Schwerpunktprogramm “Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme.”.
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Beyn, WJ., Stiefenhofer, M. A Direct Approach to Homoclinic Orbits in the Fast Dynamics of Singularly Perturbed Systems. Journal of Dynamics and Differential Equations 11, 671–709 (1999). https://doi.org/10.1023/A:1022663512855
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DOI: https://doi.org/10.1023/A:1022663512855