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Linear Nonautonomous Systems of Differential Equations with a Quadratic Integral

Linear Nonautonomous Systems of Differential Equations with a Quadratic Integral We consider nonautonomous linear systems of differential equations admittinga time-dependent first integral that is a quadratic form. The duality between mutually adjointlinear systems with quadratic integrals is established. Conditions for the spectrum of such linearsystems to be symmetric about zero are indicated. We prove that a linear system is stable if andonly if it admits a first integral that is a positive definite quadratic form. For linear systems witha quadratic integral, invariant measures whose densities are positive functions of time are studied.An explicit form of a series of quadratic integrals is specified if one of them is known. It is shownthat the degree of instability of a regular linear system (the number, counting multiplicities, ofpositive points in the spectrum) is at most the maximum of the indices of inertia of a reduciblequadratic integral. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Linear Nonautonomous Systems of Differential Equations with a Quadratic Integral

Kozlov, V. V.
Differential Equations , Volume 57 (2) – Mar 19, 2021
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References (11)

Publisher
Springer Journals
Copyright
Copyright © Pleiades Publishing, Ltd. 2021
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1134/S0012266121020063
Publisher site
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Abstract

We consider nonautonomous linear systems of differential equations admittinga time-dependent first integral that is a quadratic form. The duality between mutually adjointlinear systems with quadratic integrals is established. Conditions for the spectrum of such linearsystems to be symmetric about zero are indicated. We prove that a linear system is stable if andonly if it admits a first integral that is a positive definite quadratic form. For linear systems witha quadratic integral, invariant measures whose densities are positive functions of time are studied.An explicit form of a series of quadratic integrals is specified if one of them is known. It is shownthat the degree of instability of a regular linear system (the number, counting multiplicities, ofpositive points in the spectrum) is at most the maximum of the indices of inertia of a reduciblequadratic integral.

Journal

Differential Equations – Springer Journals

Published: Mar 19, 2021

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