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E-grāmata: Local Lyapunov Exponents: Sublimiting Growth Rates of Linear Random Differential Equations

  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 1963
  • Izdošanas datums: 17-Dec-2008
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783540859642
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Local Lyapunov Exponents: Sublimiting Growth Rates of Linear Random Differential EquationsPalielināt
  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 1963
  • Izdošanas datums: 17-Dec-2008
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783540859642
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Establishing a new concept of local Lyapunov exponents the author brings together two separate theories, namely Lyapunov exponents and the theory of large deviations.

Specifically, a linear differential system is considered which is controlled by a stochastic process that during a suitable noise-intensity-dependent time is trapped near one of its so-called metastable states. The local Lyapunov exponent is then introduced as the exponential growth rate of the linear system on this time scale. Unlike classical Lyapunov exponents, which involve a limit as time increases to infinity in a fixed system, here the system itself changes as the noise intensity converges, too.



Establishing a new concept of local Lyapunov exponents, this volume brings together two separate theories, namely Lyapunov exponents and the theory of large deviations.

Recenzijas

From the reviews:

These lecture notes originate from the author's dissertation and provide a self-contained introduction to the notion of local Lyapunov exponents. provide a beautiful exposition to this partially subtle subject for readers acquainted with the theory of stochastic differential equations. Great qualities of this book are also the ample bibliography giving a representative state of the large literature in this field and the great amount of instructively worked out examples. The composition of the text is throughout clear, carefully thought through and harmonic. (Michael Högele, Zentralblatt MATH, Vol. 1178, 2010)

Introduction 1(8)
Linear differential systems with parameter excitation
9(44)
The model
10(2)
Spherical coordinates for linear systems
12(8)
The Multiplicative Ergodic Theorem: Lyapunov exponents
20(8)
The deterministic case: Lyapunov exponents for asymptotically constant linear systems
28(16)
Sample systems
44(9)
Locality and time scales of the underlying non-degenerate stochastic system: Freidlin-Wentzell theory
53(72)
Preliminaries and assumptions
55(5)
The limiting distribution (stationary measure)
60(8)
The large deviations principle
68(4)
Exit probabilities for non-degenerate systems
72(19)
Sublimiting distributions: Metastability and quasi-deterministic behavior
91(17)
Sample systems
108(17)
Exit probabilities for degenerate systems
125(18)
Exit probabilities for degenerate systems depending on a small parameter
126(14)
Uniform consequence for the exit probability
140(3)
Local Lyapunov exponents
143(88)
Local Lyapunov exponents: upper and lower bound
144(12)
The local growth rate of the determinant
156(1)
Local Lyapunov exponents in the diagonal case
157(20)
Local Lyapunov exponents in the two-dimensional, general case
177(49)
Qualitative theory of nonlinear real noise systems on time scales
178(8)
The local Lyapunov exponent
186(40)
Concluding remarks
226(5)
Notations 231(8)
Bibliography 239(14)
Index 253
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