Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations

  • Vu Hoang Linh
  • Volker MehrmannEmail author


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. It is also discussed how numerical methods for computing the spectral intervals associated with Lyapunov and Sacker-Sell (exponential dichotomy) can be extended from those methods proposed for ODEs. Some numerical examples are presented to illustrate the theoretical results.


Differential-algebraic equations Strangeness index Lyapunov exponent Bohl exponent Sacker-Sell spectrum Exponential dichotomy Spectral interval Smooth QR factorization Continuous QR algorithm Discrete QR algorithm Kinematic equivalence Steklov function 

Mathematics Subject Classifications

65L07 65L80 34D08 34D09 


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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Faculty of Mathematics, Mechanics and InformaticsVietnam National UniversityHanoiVietnam
  2. 2.Institut für Mathematik, MA 4-5Technische Universität BerlinBerlinGermany

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