Transient spectral theory, stable and unstable cones and Gershgorin's theorem for finite-time differential equations
Resource
Journal of Differential Equations, 250(11), 4177-4199
Journal
Journal of Differential Equations
Journal Volume
250
Journal Issue
11
Pages
4177-4199
Date Issued
2011
Date
2011
Author(s)
Doan, T.S.
Palmer, K.
Siegmund, S.
Abstract
Dynamical behaviour on a compact (finite-time) interval is called monotone-hyperbolic or M-hyperbolic if there exists an invariant splitting consisting of solutions with monotonically decreasing and increasing norms, respectively. This finite-time hyperbolicity notion depends on the norm. For arbitrary norms we prove a spectral theorem based on M-hyperbolicity and extend Gershgorin's circle theorem to this type of finite-time spectrum. Similarly to stable and unstable manifolds, we characterize M-hyperbolicity by means of existence of stable and unstable cones. These cones can be explicitly computed for D-hyperbolic systems with norms induced by symmetric positive definite matrices and also for row diagonally dominant systems with the sup-norm, thus providing sufficient and computable conditions for M-hyperbolicity. © 2011 Elsevier Inc.
Type
journal article
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