Skip to main content Accessibility help
×
Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-13T15:44:30.958Z Has data issue: false hasContentIssue false

Introduction

Published online by Cambridge University Press:  05 May 2013

Luis Barreira
Affiliation:
Instituto Superior Técnico, Lisboa
Yakov Pesin
Affiliation:
Pennsylvania State University
Get access

Summary

The goal of this book is to present smooth ergodic theory from a contemporary point of view. Among other things this theory provides a rigorous mathematical foundation for the phenomenon known as deterministic chaos – a term coined by Yorke – the appearance of highly irregular, unpredictable,“chaotic” motions in pure deterministic dynamical systems. The main idea beyond this phenomenon is that one can deduce a sufficiently complete description of topological and ergodic properties of the system from relatively weak requirements on its local behavior, known as nonuniform hyperbolicity conditions: the reason this theory is also called nonuniform hyperbolicity theory.

It originated in the seminal works of Lyapunov [134] and Perron [164] on stability of solutions of ordinary differential equations. To determine whether a given solution is stable one proceeds as follows. First, the equation is linearized along the solution and then the stability of the zero solution of the corresponding nonautonomous linear differential equation is examined. There are several methods (due to Hadamard [79], Perron [165], Fenichel [70], and Irwin [92]) aimed at exhibiting stability of solutions via certain information on the linear system. The approach by Lyapunov uses a special real-valued function on the space of solutions of the linear system known as the Lyapunov exponent. It measures in the logarithmic scale the rate of convergence of solutions so that the zero solution is asymptotically exponentially stable along any subspace where the Lyapunov exponent is negative.

Type
Chapter
Information
Nonuniform Hyperbolicity
Dynamics of Systems with Nonzero Lyapunov Exponents
, pp. 1 - 6
Publisher: Cambridge University Press
Print publication year: 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Introduction
  • Luis Barreira, Instituto Superior Técnico, Lisboa, Yakov Pesin, Pennsylvania State University
  • Book: Nonuniform Hyperbolicity
  • Online publication: 05 May 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781107326026.002
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • Introduction
  • Luis Barreira, Instituto Superior Técnico, Lisboa, Yakov Pesin, Pennsylvania State University
  • Book: Nonuniform Hyperbolicity
  • Online publication: 05 May 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781107326026.002
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Introduction
  • Luis Barreira, Instituto Superior Técnico, Lisboa, Yakov Pesin, Pennsylvania State University
  • Book: Nonuniform Hyperbolicity
  • Online publication: 05 May 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781107326026.002
Available formats
×