Abstract
Along given manifolds of equilibria, bifurcations without parameters display a surprisingly rich and intricate structure of heteroclinic connections. Although manifolds of equilibria appear to be a rather degenerate feature of a vector field, the large variety of applications exhibiting this structure requires a systematic analysis of the emerging bifurcation problems. Techniques including center manifolds, normal forms and blow-up methods are indispensable for the theory.
Keywords
- Vector Field
- Bifurcation Point
- Bifurcation Theory
- Heteroclinic Orbit
- Heteroclinic Connection
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Chossat, P., Lauterbach, R.: Methods in equivariant bifurcations and dynamical systems. Advanced Series in Nonlinear Dynamics, vol. 15. World Scientific, Singapore (2000)
Kosiuk, I., Szmolyan, P.: Scaling in singular perturbation problems: blowing up a relaxation oscillator. SIAM J. Appl. Dyn. Syst. 10(4), 1307–1343 (2011)
Krupa, M., Szmolyan, P.: Extending geometric singular perturbation theory to nonhyperbolic points — fold and canard points in two dimensions. SIAM J. Math. Anal. 33(2), 286–314 (2001)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Liebscher, S. (2015). Summary and Outlook. In: Bifurcation without Parameters. Lecture Notes in Mathematics, vol 2117. Springer, Cham. https://doi.org/10.1007/978-3-319-10777-6_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-10777-6_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10776-9
Online ISBN: 978-3-319-10777-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
