title: Invariant manifolds and global bifurcations.
abstract: Invariant manifolds are key objects in describing how trajectories partition the phase spaces of a dynamical system. Examples include stable, unstable, and center manifolds of equilibria and periodic orbits, quasiperiodic invariant tori, and slow manifolds of systems with multiple timescales. Changes in these objects and their intersections with variation of system parameters give rise to global bifurcations. Bifurcation manifolds in the parameter spaces of multi-parameter families of dynamical systems also play a prominent role in dynamical systems theory. Much progress has been made in developing theory and computational methods for invariant manifolds during the past 25 years. This article highlights some of these achievements and remaining open problems.
authors.affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853, USA.
authors.forename: John
authors.initials: J
authors.lastname: Guckenheimer
authors.affiliation: Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand.
authors.forename: Bernd
authors.initials: B
authors.lastname: Krauskopf
authors.affiliation: Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand.
authors.forename: Hinke M
authors.initials: HM
authors.lastname: Osinga
authors.affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912, USA.
authors.forename: Björn
authors.initials: B
authors.lastname: Sandstede