×

Compound matrices and ordinary differential equations. (English) Zbl 0725.34049

A survey is given of a connection between compound matrices and ordinary differential equations. A typical result for linear systems is the following. If the n-th order differential equation \(x'=A(t)x\) is uniformly stable, then a necessary and sufficient condition that the equation has an \((n-k+1)\)-dimensional set of solutions satisfying \(\lim_{t\to \infty}x(t)=0\) is that \(y'=A^{[k]}(t)y\) should be asymptotically stable. For nonlinear autonomous systems, a criterion for orbital asymptotic stability of a closed trajectory given by Poincaré in two dimensions is extended to systems of any finite dimension. A criterion of Bendixson for the nonexistence of periodic solutions of a two dimensional system is also extended to higher dimensions.
Reviewer: P.Smith (Keele)

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. C. Aitken, Determinants and Matrices, Oliver and Boyd, Edinburgh, 1956.
[2] R. Bellman, Introduction to Matrix Analysis, McGraw-Hill, New York, 1960. · Zbl 0124.01001
[3] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Health, Boston, 1965. · Zbl 0154.09301
[4] J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), 263-321. JSTOR: · Zbl 0001.14102
[5] M Fiedler, Additive compound matrices and inequality for eigenvalues of stochastic matrices, Czechoslovak Math. J. 24 (99) (1974), 392-402. · Zbl 0345.15013
[6] F. R. Gantmacher, The Theory of Matrices, Chelsea Publ. Co., New York, 1959. · Zbl 0085.01001
[7] W. Hahn, Stability of Motion, Springer-Verlag, New York, 1967. · Zbl 0189.38503
[8] J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, 1969. · Zbl 0186.40901
[9] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964; Birkhäuser, Boston, 1982. · Zbl 0125.32102
[10] A. S. Householder, The Theory of Matrices in Numerical Analysis, Blaisdell, New York, 1964. · Zbl 0161.12101
[11] P. Lancaster and M. Tismenetsky, The Theory of Matrices, Second Edition with Applications, Academic Press, Orlando, 1985. · Zbl 0558.15001
[12] D. London, On derivations arising in differential equations, Linear and Multilinear Algebra 4 (1976), 179 -189. · Zbl 0358.15011
[13] M. Marcus and H. Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Boston, 1964. · Zbl 0126.02404
[14] A. W. Marshall and I. Olkin, Inequalities: theory of majorization and its applications, Academic Press, New York, 1979. · Zbl 0437.26007
[15] J. Mikusiński, Sur l’equation \(x^(n) +A(t)x=0\), Ann. Polon. Math. 1 (1955), 207-221. · Zbl 0064.33104
[16] T. Muir, The theory of determinants in the historical order of development, Macmillan, London, 1906. · JFM 37.0181.02
[17] J. S. Muldowney, On the dimension of the zero or infinity tending sets for linear differential equations, Proc. Amer. Math. Soc. 83 (1981), 705-709. JSTOR: · Zbl 0484.34003
[18] ——–, Dichotomies and asymptotic behaviour for linear differential systems, Trans. Amer. Math. Soc. 283 (1984), 465-484. JSTOR: · Zbl 0559.34049
[19] Z. Nehari, Disconjugate linear differential operators, Trans. Amer. Math. Soc. 129 (1967), 500-516. JSTOR: · Zbl 0183.09101
[20] G. B. Price, Some identities in the theory of determinants, Amer. Math. Monthly 54 (1947), 75-90. JSTOR: · Zbl 0029.00210
[21] B. Schwarz, Totally positive differential systems, Pacific J. Math. 32 (1970), 203-229. · Zbl 0193.04501
[22] R. A. Smith, An index theorem and Bendixson’s negative criterion for certain differential equations of higher dimension, Proc. Roy. Soc. Edinburgh 91A (1981), 63-77. · Zbl 0499.34026
[23] ——–, Some applications of Hausdorff dimension inequalities for ordinary differential equations, Proc. Roy. Soc. Edinburgh 104A (1986), 235-259. · Zbl 0622.34040
[24] J. H. M. Wedderburn, Lectures on matrices, Amer. Math. Soc., New York, 1934. · Zbl 0121.26101
[25] H. Wielandt, Topics in the analytic theory of matrices, Lecture notes prepared by R.R. Meyer, University of Wisconsin, Madison, 1967. · Zbl 0178.02104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.