Abstract
We consider singularly perturbed systems\(\xi = f(\xi ,\eta ,\varepsilon ),\dot \eta = \varepsilon g(\xi ,\eta ,\varepsilon )\), such thatξ=f(ξ, αo, 0). αo∃ℝm, has a heteroclinic orbitu(t). We construct a bifurcation functionG(α, ɛ) such that the singular system has a heteroclinic orbit if and only ifG(α, ɛ)=0 has a solutionα=α(ɛ). We also apply this result to recover some theorems that have been proved using different approaches.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Battelli, F., and Lazzari, C. (1993). Heteroclinic orbits in systems with slowly varying coefficients.J. Diff. Eq. 105, 1–29.
Battelli, F., and Palmer, K. J. (1993). Chaos in the Duffing equation.J. Diff. Eq. 101, 276–301.
Cherry, T. M. (1968). Asymptotic solutions of analytical hamiltonian systems.J. Diff. Eq. 4, 142–159.
Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations.J. Diff. Eq. 31, 53–98.
Kurland, H., and Levi, M. (1988). Transversal heteroclinic intersections in slowly varying systems. In Salam, F. A., and Levi, M. (eds.),Dynamical Systems Approaches to Nonlinear Problems in Systems and Circuits, SIAM, pp. 29–38.
Lin, X. B. (1989). Shadowing lemma and singularly perturbed boundary value problems.SIAM J. Appl. Math. 49, 26–54.
Palmer, K. J. (1986). Transverse heteroclinic orbits and Cherry's example of a nonintegrable Hamiltonian system.J. Diff. Eq. 65, 321–360.
Palmer, K. J. (1990). Existence of a tranversal homoclinic point in a degenerate case.Rocky Mt. J. Math. 20, 1099–1118.
Sakamoto, K. (1990). Invariant manifolds in singular perturbation problems for ordinary differential equations.Proc. Roy. Soc. Edinburgh Sect. A 116, 45–78.
Szmolyan, P. (1991). Transversal heteroclinic and homoclinic orbits in singular perturbation problems.J. Diff. Eq. 92, 252–281.
Wiggins, S. (1988).Global Bifurcation and Chaos, Springer-Verlag, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Battelli, F. Heteroclinic orbits in singular systems: A unifying approach. J Dyn Diff Equat 6, 147–173 (1994). https://doi.org/10.1007/BF02219191
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02219191