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Invariant manifolds and global bifurcations.
John Guckenheimer1, Bernd Krauskopf2, Hinke M Osinga2
1Department of Mathematics, Cornell University, Ithaca, New York 14853, USA.
Chaos (Woodbury, N.Y.)
|October 3, 2015
Summary
Invariant manifolds are crucial for understanding dynamical systems, partitioning phase spaces and leading to global bifurcations. Recent advancements have improved their theory and computation, with ongoing research addressing open problems.
Area of Science:
- Mathematics
- Dynamical Systems Theory
- Differential Equations
Background:
- Invariant manifolds are fundamental to understanding the geometric structure of dynamical systems.
- They include stable, unstable, center manifolds, invariant tori, and slow manifolds.
- Their intersections and parameter variations lead to global bifurcations.
Purpose of the Study:
- To review recent progress in the theory and computational methods for invariant manifolds.
- To highlight key achievements in the field over the past 25 years.
- To identify remaining open problems in invariant manifold research.
Main Methods:
- Theoretical analysis of dynamical systems.
- Development of computational techniques for manifold computation.
- Review of existing literature and research findings.
Main Results:
- Significant advancements in the theory of invariant manifolds.
- Improved computational methods for identifying and analyzing invariant manifolds.
- Identification of critical areas for future research in dynamical systems.
Conclusions:
- The theory and computation of invariant manifolds have seen substantial progress.
- Invariant manifolds remain central to understanding global bifurcations and system dynamics.
- Further research is needed to address outstanding challenges in the field.
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