Skip to main content

Lipschitz stable manifolds

  • Chapter
  • 1519 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1926)

We want to construct stable and unstable invariant manifolds without assuming the existence of a uniform exponential dichotomy for the linear variational equation. Our main objective is to describe the weakest possible setting under which one can construct the invariant manifolds.We still require some amount of hyperbolicity. Namely, we show that under fairly general assumptions the generalized notion of nonuniform exponential dichotomy allows us to establish the existence of stable and unstable invariant manifolds. In this chapter we only consider “Lipschitz manifolds”, that is, graphs of Lipschitz functions. We refer to Chapters 5 and 6 for the existence of smooth invariant manifolds (respectively in Rn and in arbitrary Banach spaces), under slightly stronger assumptions. We follow closely [12], although now considering the general case when the stable and unstable subspaces may depend on the time t. Lipschitz center manifolds were obtained with a similar approach in [8]; we refer to Chapter 8 for the construction of smooth center manifolds.

Keywords

  • Unique Function
  • Lipschitz Function
  • Invariant Manifold
  • Unstable Manifold
  • Stable Manifold

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

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   49.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Rights and permissions

Reprints and Permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

(2008). Lipschitz stable manifolds. In: Stability of Nonautonomous Differential Equations. Lecture Notes in Mathematics, vol 1926. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74775-8_4

Download citation