Liapunov exponents for higher-order linear differential equations whose characteristic equations have variable real roots

Michael I. Gil

Liapunov exponents for higher-order linear differential equations whose characteristic equations have variable real roots

Michael I. Gil

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We consider the linear differential equation Σ_k=0 ⁿaak(t)x(n-k)(t) = 0 t ≥ 0, n≥2, where a₀(t) ≡ 1, ak(t) are continuous bounded functions. Assuming that all the roots of the polynomial zⁿ + a₁ (t)z^n-1 + ⋯ + a _n(t) are real and satisfy the inequality r_k(t) < γ for t ≥ 0 and k = 1,..., n, we prove that the solutions of the above equation satisfy |x(i)| ≤ const e^γt for t ≥ 0.

Original language	English
Pages (from-to)	1-6
Number of pages	6
Journal	Electronic Journal of Differential Equations
Volume	2008
State	Published - 15 Apr 2008

Keywords

Exponential stability
Liapunov exponents
Linear differential equations

ASJC Scopus subject areas

Analysis

Cite this

@article{a012962fd26d4b9bb015a14156685f77,

title = "Liapunov exponents for higher-order linear differential equations whose characteristic equations have variable real roots",

abstract = "We consider the linear differential equation Σk=0 naak(t)x(n-k)(t) = 0 t ≥ 0, n≥2, where a0(t) ≡ 1, ak(t) are continuous bounded functions. Assuming that all the roots of the polynomial zn + a1 (t)zn-1 + ⋯ + a n(t) are real and satisfy the inequality rk(t) < γ for t ≥ 0 and k = 1,..., n, we prove that the solutions of the above equation satisfy |x(i)| ≤ const eγt for t ≥ 0.",

keywords = "Exponential stability, Liapunov exponents, Linear differential equations",

author = "Gil, \{Michael I.\}",

note = "Funding Information: Generous financial support from the Swedish Natural Science Research Council (Vetenskapsr{\aa}det), the Swedish Strategic Research Foundation (Stiftelsen f{\"o}r Strategisk Forskning), European Union Biotechnology Program (LSHB-CT-2004-005276, Contract No. 005276), and Philip Morris USA, Inc. is gratefully acknowledged. The computational resources have been granted by the Swedish National Allocation Committee for the High Performance Computing and provided by the National Supercomputer Center (Link{\"o}ping, Sweden) for what the authors are thankful for.",

year = "2008",

month = apr,

day = "15",

language = "English",

volume = "2008",

pages = "1--6",

journal = "Electronic Journal of Differential Equations",

issn = "1072-6691",

publisher = "Texas State University - San Marcos",

}

TY - JOUR

T1 - Liapunov exponents for higher-order linear differential equations whose characteristic equations have variable real roots

AU - Gil, Michael I.

N1 - Funding Information: Generous financial support from the Swedish Natural Science Research Council (Vetenskapsrådet), the Swedish Strategic Research Foundation (Stiftelsen för Strategisk Forskning), European Union Biotechnology Program (LSHB-CT-2004-005276, Contract No. 005276), and Philip Morris USA, Inc. is gratefully acknowledged. The computational resources have been granted by the Swedish National Allocation Committee for the High Performance Computing and provided by the National Supercomputer Center (Linköping, Sweden) for what the authors are thankful for.

PY - 2008/4/15

Y1 - 2008/4/15

N2 - We consider the linear differential equation Σk=0 naak(t)x(n-k)(t) = 0 t ≥ 0, n≥2, where a0(t) ≡ 1, ak(t) are continuous bounded functions. Assuming that all the roots of the polynomial zn + a1 (t)zn-1 + ⋯ + a n(t) are real and satisfy the inequality rk(t) < γ for t ≥ 0 and k = 1,..., n, we prove that the solutions of the above equation satisfy |x(i)| ≤ const eγt for t ≥ 0.

AB - We consider the linear differential equation Σk=0 naak(t)x(n-k)(t) = 0 t ≥ 0, n≥2, where a0(t) ≡ 1, ak(t) are continuous bounded functions. Assuming that all the roots of the polynomial zn + a1 (t)zn-1 + ⋯ + a n(t) are real and satisfy the inequality rk(t) < γ for t ≥ 0 and k = 1,..., n, we prove that the solutions of the above equation satisfy |x(i)| ≤ const eγt for t ≥ 0.

KW - Exponential stability

KW - Liapunov exponents

KW - Linear differential equations

UR - https://www.scopus.com/pages/publications/42449152487

M3 - Article

AN - SCOPUS:42449152487

SN - 1072-6691

VL - 2008

SP - 1

EP - 6

JO - Electronic Journal of Differential Equations

JF - Electronic Journal of Differential Equations

ER -

Liapunov exponents for higher-order linear differential equations whose characteristic equations have variable real roots

Abstract

Keywords

ASJC Scopus subject areas

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