Refine
Language
- English (3)
Keywords
- Lyapunov exponent (3)
- Sacker-Sell spectrum (3)
- exponential dichotomy (3)
- spectral interval (3)
- Bohl exponent (2)
- Steklov function (2)
- differential-algebraic equation (2)
- index (2)
- strangeness (2)
- QR algorithm (1)
Project
- D1 (3)
Application Area
- D (3)
In this paper, we propose and investigate numerical methods based on QR factorization for computing all or some Lyapunov or Sacker-Sell spectral intervals for
linear differential-algebraic equations.
Furthermore, a perturbation and error analysis for these methods is presented. We
investigate how errors in the data and in the numerical integration affect the
accuracy of the approximate spectral intervals. Although we need to integrate
numerically some differential-algebraic systems on usually very long
time-intervals, under certain assumptions, it is shown that the error of the
computed spectral intervals can be controlled by the local error of numerical
integration and the error in solving the algebraic constraint.
Some numerical examples are presented to illustrate the theoretical results.
This paper is devoted to the numerical approximation of Lyapunov and Sacker-Sell spectral intervals for linear differential-algebraic equations (DAEs). The spectral analysis for DAEs is improved and the concepts of leading directions and solution subspaces associated with spectral intervals are extended to DAEs. Numerical methods
based on smooth singular value decompositions are introduced for computing all or only some spectral intervals and their associated leading directions. The numerical algorithms as well as implementation issues are discussed in detail and numerical examples are presented to illustrate the theoretical results.
Lyapunov and exponential dichotomy spectral theory is extended
from ordinary differential equations (ODEs) to nonautonomous
differential-algebraic equations (DAEs). By using orthogonal
changes of variables, the original DAE system is transformed into
appropriate condensed forms, for which concepts such as Lyapunov
exponents, Bohl exponents, exponential dichotomy and spectral
intervals of various kinds can be analyzed via the resulting
underlying ODE. Some essential differences between the spectral
theory for ODEs and that for DAEs are pointed out. Numerical
methods for computing the spectral intervals associated with
Lyapunov and Sacker-Sell (exponential dichotomy) spectra are
derived by modifying and extending those methods proposed for ODEs. Perturbation theory and error analysis are discussed, as
well. Finally, some numerical examples are presented to illustrate
the theoretical results and the properties of the numerical
methods.