A unified approach to finite-time hyperbolicity which extends finite-time Lyapunov exponents

A hyperbolicity notion for linear differential equations x˙=A(t)x, t∈[t−,t+], is defined which unifies different existing notions like finite-time Lyapunov exponents (Haller, 2001, [13], Shadden et al., 2005, [24]), uniform or M-hyperbolicity (Haller, 2001, [13], Berger et al., 2009, [6]) and (t−,(t...

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Bibliographic Details
Published in:Journal of Differential Equations 2012-05, Vol.252 (10), p.5535-5554
Main Authors: Doan, T.S., Karrasch, D., Nguyen, T.Y., Siegmund, S.
Format: Article
Language:English
Subjects:
Finite-time hyperbolicity
Finite-time Lyapunov exponents
Spectral theorem
Transient dynamics
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Summary:A hyperbolicity notion for linear differential equations x˙=A(t)x, t∈[t−,t+], is defined which unifies different existing notions like finite-time Lyapunov exponents (Haller, 2001, [13], Shadden et al., 2005, [24]), uniform or M-hyperbolicity (Haller, 2001, [13], Berger et al., 2009, [6]) and (t−,(t+−t−))-dichotomy (Rasmussen, 2010, [21]). Its relation to the dichotomy spectrum (Sacker and Sell, 1978, [23], Siegmund, 2002, [26]), D-hyperbolicity (Berger et al., 2009, [6]) and real parts of the eigenvalues (in case A is constant) is described. We prove a spectral theorem and provide an approximation result for the spectral intervals.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2012.02.002