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Möbius transformations in stability theory

Published online by Cambridge University Press:  24 October 2008

Russell A. Smith
Russell A. Smith
Affiliation:
University of Durham
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Extract

1. Introduction: Consider the system of ordinary differential equations

where the unknown x(t) is a complex m-vector, t is a real variable, D is the operator d/dt and a0, …, an are complex m × m matrices whose elements are continuous functions of t, x, Dx, …, Dn−1x. Furthermore, det a0 ╪ 0. In the special case when a0, …, an are constant matrices the trivial solution x = 0 is asymptotically stable if and only if all the roots of the characteristic equation det f(ζ) = 0 have negative real parts, where

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 68 , Issue 1 , July 1970 , pp. 143 - 151
Copyright
Copyright © Cambridge Philosophical Society 1970

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References

REFERENCES

(1)Hahn, W.Theory and application of Liapunov's direct method (Prentice-Hall; Englewood Cliffs, 1963).Google Scholar
(2)Smith, R. A.Stability of certain systems of second order differential equations. Ann. Mat. Pura Appl. (4) 82 (1969), 257–266.CrossRefGoogle Scholar
(3)Starzhinskii, V. M.Sufficient conditions for stability of a mechanical system with one degree of freedom (Russian). Prikl. Mat. Meh. 16 (1952), 369–374.Google Scholar
(4)Starzhinskii, V. M.On stability of unsteady motions in one case (Russian). Prikl. Mat. Meh. 16 (1952), 500–504.Google Scholar
(5)Starzhinskii, V. M.On stability of unsteady motions in a special case (Russian). Prikl. Mat. Meh. 19 (1955), 471–480.Google Scholar
(6)Taussky, O.Matrices C with Cn → 0. J. Algebra 1 (1964), 5–10.CrossRefGoogle Scholar