Abstract
We consider a singularly perturbed system depending on two parameters with two (possibly the same) normally hyperbolic center manifolds. We assume that the unperturbed system has an orbit that connects a hyperbolic fixed point on one center manifold to a hyperbolic fixed point on the other. Then we prove some old and new results concerning the persistence of these connecting orbits and apply the results to find examples of systems in dimensions greater than three that possess Sil’nikov saddle-focus homoclinic orbits.
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References
F. Battelli and K. J. Palmer, “Singular perturbations, transversality, and Sil’nikov saddle focus homoclinic orbits,” J. Dynam. Different. Equat., 15, 357–425 (2003).
P. Szmolyan, “Transversal heteroclinic and homoclinic orbits in singular perturbation problems,” J. Different. Equat., 92, 252–281 (1991).
W.-J. Beyn and M. Stiefenhofer, “A direct approach to homoclinic orbits in the fast dynamics of singularly perturbed systems,” J. Dynam. Different. Equat., 99, 671–709 (1999).
F. Battelli and K. J. Palmer, “Heteroclinic connections in singularly perturbed systems,” Disc. Cont. Dynam. Syst. (to appear).
F. Battelli and K. J. Palmer, “Transverse intersection of invariant manifolds in singular systems,” J. Different. Equat., 177, 77–120 (2001).
L. P. Sil’nikov, “A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type,” Mat. Sb., 10, 91–102 (1970).
B. Deng, “On Sil’nikov’s homoclinic-saddle-focus theorem,” J. Different. Equat., 102, 305–329 (1993).
S. P. Hastings, “Single and multiple pulse waves for FitzHugh Nagumo equation,” SIAM J. Appl. Math., 42, 247–260 (1982).
B. Deng and G. Hines, “Food chain chaos due to Sil’nikov’s orbits,” Chaos, 12, 533–538 (2002).
Z. C. Feng and S. Wiggins, “On the existence of chaos in a class of two degree of freedom, damped, parametrically forced mechanical systems with broken O(2) symmetry,” Z. Angew. Math. Phys., 44, 201–248 (1993).
J. A. Rodriguez, “Bifurcation to homoclinic connections of the focus-saddle type,” Arch. Ration. Mech. Anal., 93, 81–90 (1986).
F. Battelli and K. J. Palmer, “A remark about Sil’nikov saddle-focus homoclinic orbits” (to appear).
F. Fenichel, “Geometric singular perturbation theory for ordinary differential equations,” J. Different. Equat., 31, 53–98 (1979).
K. Sakamoto, “Invariant manifolds in singular perturbation problems for ordinary differential equations,” Proc. Roy. Soc. Edinburgh A, 116, 45–78 (1990).
F. Battelli and M. Fečkan, “Global centre manifolds in singular systems,” Nonlinear Different. Equat. Appl., 3, 19–34 (1996).
R. A. Johnson and G. R. Sell, “Smoothness of spectral subbundles and reducibility of quasiperiodic linear differential systems,” J. Different. Equat., 41, 262–288 (1981).
J. K. Hale, “Introduction to dynamic bifurcation,” Bifurcation Theory Appl. Lect. Notes Math., 1057, 106–151 (1984).
K. J. Palmer, “Transverse heteroclinic orbits and Cherry’s example of a nonintegrable Hamiltonian system,” J. Different. Equat., 65, 321–360 (1986).
K. J. Palmer, “Existence of a transverse homoclinic point in a degenerate case,” Rocky Mountain J. Math., 20, 1099–1118 (1990).
K. J. Palmer, “Exponential dichotomies and transversal homoclinic points,” J. Different. Equat., 55, 225–256 (1984).
H. Kokubu, K. Mischaikow, and H. Oka, “Existence of infinitely many connecting orbits in a singularly perturbed ordinary differential equation,” Nonlinearity, 9, 1263–1280 (1996).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 28–55, January, 2008.
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Battelli, F., Palmer, K.J. Connections to fixed points and Sil’nikov saddle-focus homoclinic orbits in singularly perturbed systems. Ukr Math J 60, 29–58 (2008). https://doi.org/10.1007/s11253-008-0040-9
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DOI: https://doi.org/10.1007/s11253-008-0040-9