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Stability of solutions in different variables

Stability of solutions in different variables ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 11, pp. 1505–1509.  c Pleiades Publishing, Ltd., 2007. Original Russian Text  c A.V. Kavinov, A.P. Krishchenko, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 11, pp. 1470–1473. ORDINARY DIFFERENTIAL EQUATIONS Stability of Solutions in Different Variables A. V. Kavinov and A. P. Krishchenko Moscow State Technical University, Moscow, Russia Institute for System Analysis, Russian Academy of Sciences, Moscow, Russia Received July 4, 2007 DOI: 10.1134/S0012266107110043 The problem of stability in various variables has been known since the nineteenth century. Lyapunov [1, pp. 12–13] noted that the stability of a solution of a system in some variables does not imply the stability of the same solution in other variables in general. The following simplest example illustrates this fact. In the theory of linear systems, it is known that all solutions of the differential equation x ˙ = x (1) are unstable. By performing the change of variables y =arctan x, x =tan y, we obtain the equivalent equation x ˙ 1 y˙ = =tan y cos y =sin y cos y = sin 2y (2) 1+ x 2 provided that y ∈ (−π/2; π/2). By integrating Eq. (2), we http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Stability of solutions in different variables

Kavinov, A.; Krishchenko, A.
Differential Equations , Volume 43 (11) – Mar 24, 2007
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References (4)

Publisher
Springer Journals
Copyright
Copyright © 2007 by Pleiades Publishing, Ltd.
Subject
Mathematics; Ordinary Differential Equations; Partial Differential Equations; Difference and Functional Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1134/S0012266107110043
Publisher site
See Article on Publisher Site

Abstract

ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 11, pp. 1505–1509.  c Pleiades Publishing, Ltd., 2007. Original Russian Text  c A.V. Kavinov, A.P. Krishchenko, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 11, pp. 1470–1473. ORDINARY DIFFERENTIAL EQUATIONS Stability of Solutions in Different Variables A. V. Kavinov and A. P. Krishchenko Moscow State Technical University, Moscow, Russia Institute for System Analysis, Russian Academy of Sciences, Moscow, Russia Received July 4, 2007 DOI: 10.1134/S0012266107110043 The problem of stability in various variables has been known since the nineteenth century. Lyapunov [1, pp. 12–13] noted that the stability of a solution of a system in some variables does not imply the stability of the same solution in other variables in general. The following simplest example illustrates this fact. In the theory of linear systems, it is known that all solutions of the differential equation x ˙ = x (1) are unstable. By performing the change of variables y =arctan x, x =tan y, we obtain the equivalent equation x ˙ 1 y˙ = =tan y cos y =sin y cos y = sin 2y (2) 1+ x 2 provided that y ∈ (−π/2; π/2). By integrating Eq. (2), we

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Differential Equations – Springer Journals

Published: Mar 24, 2007

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