Abstract
Center manifolds facilitate the reduction of the dimension of a bifurcation problem to the necessary minimum. The local center manifold of an equilibrium, i.e. the bifurcation point, is a smooth manifold tangential to the center eigenspace of that equilibrium. The center eigenspace is the generalized eigenspace to all purely imaginary (or zero) eigenvalues of the linearization of the vector field at the equilibrium. The local center manifold contains all bounded solution in a small neighborhood, in particular all equilibria, all periodic orbits, and all connecting (heteroclinic) orbits of equilibria.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Arnol’d, V.: Dynamical systems V. Bifurcation theorie and catastrophe theory. Enc. Math. Sciences, vol. 5. Springer, Berlin (1994)
Dumortier, F., Roussarie, R.: Geometric singular perturbation theory beyond normal hyperbolicity. In: Jones, C.K.R.T. et al. (eds.) Multiple-Time-Scale Dynamical Systems. Proceedings of the IMA Workshop, Minneapolis, 1997–1998, IMA Vol. Math. Appl., vol. 122, pp. 29–63. Springer, New York (2001)
Fiedler, B., Liebscher, S.: Takens-Bogdanov bifurcations without parameters, and oscillatory shock profiles. In: Broer, H., Krauskopf, B., Vegter, G. (eds.) Global Analysis of Dynamical Systems, Festschrift dedicated to Floris Takens for his 60th birthday, pp. 211–259. IOP, Bristol (2001)
Hirsch, M., Pugh, C., Shub, M.: Invariant Manifolds. Lect. Notes Math., vol. 583. Springer, Berlin (1977)
Kosiuk, I., Szmolyan, P.: Scaling in singular perturbation problems: blowing up a relaxation oscillator. SIAM J. Appl. Dyn. Syst. 10(4), 1307–1343 (2011)
Murdock, J.: Normal forms and unfoldings for local dynamical systems. Monogr. in Math. Springer, New York (2003)
Vanderbauwhede, A.: Centre manifolds, normal forms and elementary bifurcations. In: Kirchgraber, U., Walther, H.O. (eds.) Dynamics Reported 2, pp. 89–169. Teubner & Wiley, Stuttgart (1989)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Liebscher, S. (2015). Methods and Concepts. In: Bifurcation without Parameters. Lecture Notes in Mathematics, vol 2117. Springer, Cham. https://doi.org/10.1007/978-3-319-10777-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-10777-6_2
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)