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Erschienen in:
Nonautonomous Linear Hamiltonian Systems Kapitel lesen Erstes Kapitel lesen

2016 | OriginalPaper | Buchkapitel

2. The Rotation Number and the Lyapunov Index for Real Nonautonomous Linear Hamiltonian Systems

verfasst von : Russell Johnson, Rafael Obaya, Sylvia Novo, Carmen Núñez, Roberta Fabbri

Erschienen in: Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control

Verlag: Springer International Publishing

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Abstract

In this chapter, two objects are introduced and analyzed, namely the rotation number and the Lyapunov index for a family of nonautonomous linear 2n-dimensional Hamiltonian systems. They depend on the choice of a fixed ergodic measure on the space Ω (whose elements determine the different systems of the family). For the rotation number, several definitions of very different nature are given, and the equivalence of all these definitions is proved. The fundamental property of continuity of the rotation number with respect to the coefficient matrix is also proved. The definition and the most basic properties of the Lyapunov index are described in the last section of the chapter, which in particular contains a proof of the upper semicontinuity of the index with respect to the coefficient matrix. In this chapter, and also in those that follow, the general analysis includes that of the dynamics given by a nonautonomous n-dimensional second order linear Schrödinger equation, which can always be written as a 2n-dimensional linear Hamiltonian system.

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Vorheriges Kapitel Nonautonomous Linear Hamiltonian Systems
Nächstes Kapitel The Floquet Coefficient for Nonautonomous Linear Hamiltonian Systems: Atkinson Problems
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Metadaten
Titel
The Rotation Number and the Lyapunov Index for Real Nonautonomous Linear Hamiltonian Systems
verfasst von
Russell Johnson
Rafael Obaya
Sylvia Novo
Carmen Núñez
Roberta Fabbri
Copyright-Jahr
2016
Buch
Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control

Print ISBN: 978-3-319-29023-2

Electronic ISBN: 978-3-319-29025-6

Copyright-Jahr: 2016

DOI
https://doi.org/10.1007/978-3-319-29025-6_2

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