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On sets of linear differential systems to which perturbed linear systems cannot be reduced

On sets of linear differential systems to which perturbed linear systems cannot be reduced The paper [2] defines the noncoinciding irreducibility sets N 2(a, σ) and N 3(a, σ), σ ∈ (0, 2a], of all n-dimensional linear differential systems with piecewise continuous coefficient matrices A(t) such that ‖A(t)‖ ≤ a < + ∞ for t ∈ [0,+∞) and there exists a linear differential system that is not Lyapunov reducible to the original system and has coefficient matrix B(t) satisfying [for the case of N 2(a, σ)] the condition $\left\| {B(t) - A(t)} \right\| \leqslant const \times e^{ - \sigma t} ,t \geqslant 0,$ or [for the case of N 3(a, σ)] the more general condition that the Lyapunov exponent of the difference B(t) − A(t) does not exceed −σ. For these sets, which are related by the obvious inclusions $N_i (a,\sigma _1 ) \supseteq N_i (a,\sigma _2 ),0 < \sigma _1 < \sigma _2 \leqslant 2a,i = 2,3,$ , we prove that (i) they strictly decrease with increasing parameter σ ∈ (0, 2a], N i (a, σ 1) ⊃ N i (a, σ 2) for σ 1 < σ 2; (ii) there is a strict inclusion N 2(a, σ) ⊂ N 3(a, σ) for all σ ∈ (0, 2a]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

On sets of linear differential systems to which perturbed linear systems cannot be reduced

Izobov, N.; Mazanik, S.
Differential Equations , Volume 47 (11) – Feb 5, 2012
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References (4)

Publisher
Springer Journals
Copyright
Copyright © 2011 by Pleiades Publishing, Ltd.
Subject
Mathematics; Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1134/S0012266111110036
Publisher site
See Article on Publisher Site

Abstract

The paper [2] defines the noncoinciding irreducibility sets N 2(a, σ) and N 3(a, σ), σ ∈ (0, 2a], of all n-dimensional linear differential systems with piecewise continuous coefficient matrices A(t) such that ‖A(t)‖ ≤ a < + ∞ for t ∈ [0,+∞) and there exists a linear differential system that is not Lyapunov reducible to the original system and has coefficient matrix B(t) satisfying [for the case of N 2(a, σ)] the condition $\left\| {B(t) - A(t)} \right\| \leqslant const \times e^{ - \sigma t} ,t \geqslant 0,$ or [for the case of N 3(a, σ)] the more general condition that the Lyapunov exponent of the difference B(t) − A(t) does not exceed −σ. For these sets, which are related by the obvious inclusions $N_i (a,\sigma _1 ) \supseteq N_i (a,\sigma _2 ),0 < \sigma _1 < \sigma _2 \leqslant 2a,i = 2,3,$ , we prove that (i) they strictly decrease with increasing parameter σ ∈ (0, 2a], N i (a, σ 1) ⊃ N i (a, σ 2) for σ 1 < σ 2; (ii) there is a strict inclusion N 2(a, σ) ⊂ N 3(a, σ) for all σ ∈ (0, 2a].

Journal

Differential Equations – Springer Journals

Published: Feb 5, 2012

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