Lyapunov and Sacker–Sell Spectral Intervals

Dieci, Luca; Vleck, Erik S. Van

doi:10.1007/s10884-006-9030-5

Cited by 65 publications

(79 citation statements)

References 17 publications

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“…We refer to [26,27,30,44] or [58] for more details on the theory for ODEs. An essential step in the computation of spectral intervals for linear DAEs of the form (1) is to first transform the system to a reduced strangeness-free form (7), which has the same solution set as (1), see [50], and then to consider the spectral results in this framework.…”

Section: Spectral Theory For Daesmentioning

confidence: 99%

“…In this section we extend the approaches that were derived for the computation of spectral intervals for ODEs in [26,27,30] to DAEs. We derive numerical methods for computing Lyapunov and Sacker-Sell spectra for DAEs of the form (9) based on smooth QR factorizations.…”

Section: Numerical Computation Of Spectral Intervals For Daesmentioning

confidence: 99%

“…Unlike the development of the analytic theory, the development of numerical methods to compute Lyapunov exponents and also other spectral intervals has only recently been studied. In a series of papers, see [23,24,26,27,29,30,31], Dieci and Van Vleck have developed algorithms for the computation of Lyapunov and Bohl exponents as well as Sacker-Sell spectral intervals. These methods have also been analyzed concerning their sensitivity under small perturbations (stability), the relationship between different spectra, the error analysis, and efficient implementation techniques.…”

Section: Introductionmentioning

confidence: 99%

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

Linh

Mehrmann

2009

J Dyn Diff Equat

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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.

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Section: Spectral Theory For Daesmentioning

confidence: 99%

Section: Numerical Computation Of Spectral Intervals For Daesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Linh

Mehrmann

2009

J Dyn Diff Equat

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“…Sacker & Sell (1978), Aulbach & Siegmund (2001), Dieci & Van Vleck (2007) also called dichotomy spectrum is for discrete time dynamical systems defined as σ ED = {γ ∈ Ê + : (17) possesses no exponential dichotomy on },…”

Section: Sacker-sell Spectrummentioning

confidence: 99%

“…Dieci & Van Vleck (2007) for continuous time systems. In discrete time, a generalization to non-invertible systems in given in Aulbach & Siegmund (2001).…”

Section: Introductionmentioning

confidence: 99%

Computing Sacker–Sell spectra in Discrete Time Dynamical Systems

Hüls¹

2010

SIAM J. Numer. Anal.

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In this paper we develop two boundary value methods for detecting SackerSell spectra in discrete time dynamical systems. The algorithms are advancements of earlier methods for computing projectors of exponential dichotomies. The first method is based on the projector residual P P − P . If this residual is large, then the difference equation has no exponential dichotomy. A second criterion for detecting Sacker-Sell spectral intervals is the norm of end points of the solution of a specific boundary value problem. Refined error estimates for the underlying approximation process are given and the resulting algorithms are applied to an example with known continuous Sacker-Sell spectrum, as well as to the variational equation along orbits of Hénon's map.

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Assessing the Roles of Fire Frequency and Precipitation in Determining Woody Plant Expansion in Central U.S. Grasslands

et al. 2017

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Woody plant expansion into grasslands and savannas is occurring and accelerating worldwide and often impacts ecosystem processes. Understanding and predicting the environmental and ecological impacts of encroachment has led to a variety of methodologies for assessing its onset, transition, and stability, generally relying on dynamical systems approaches. Here we continue this general line of investigation to facilitate the understanding of the roles of precipitation frequency and intensity and fire frequency on the conversion of grasslands to woody‐dominated systems focusing on the central United States. A low‐dimensional model with stochastic precipitation and fire disturbance is introduced to examine the complex interactions between precipitation and fire as mechanisms that may suppress or facilitate increases in woody cover. By using Lyapunov exponents, we are able to ascertain the relative control exerted on woody encroachment through these mechanisms. Our results indicate that precipitation frequency is a more important control on woody encroachment than the intensity of individual precipitation events. Fire, however, exerts a much more dominant impact on the limitation of encroachment over the range of precipitation variability considered here. These results indicate that fire management may be an effective strategy to slow the onset of woody species into grasslands. While climate change might predict a reduced potential for woody encroachment in the near future, these results indicate a reduction in woody fraction may be unlikely when considering anthropogenic fire suppression.

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Lyapunov and Sacker–Sell Spectral Intervals

Cited by 65 publications

References 17 publications

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

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

Computing Sacker–Sell spectra in Discrete Time Dynamical Systems

Assessing the Roles of Fire Frequency and Precipitation in Determining Woody Plant Expansion in Central U.S. Grasslands

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