zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 $\lim\sb{t\to \infty}x(t)=0$ is that $y'=A\sp{[k]}(t)y$ 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:
34D05Asymptotic stability of ODE
34C25Periodic solutions of ODE
WorldCat.org
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 · doi:10.2307/1989472 · http://links.jstor.org/sici?sici=0002-9947%28193101%2933%3A1%3C263%3ASOTPOP%3E2.0.CO%3B2-%23&origin=euclid
[5] M Fiedler, Additive compound matrices and inequality for eigenvalues of stochastic matrices, Czechoslovak Math. J. 24 (99) (1974), 392-402. · Zbl 0345.15013 · eudml:12802
[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 · doi:10.1080/03081087608817149
[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. · Zbl 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 · doi:10.2307/2044238 · http://links.jstor.org/sici?sici=0002-9939%28198112%2983%3A4%3C705%3AOTDOTZ%3E2.0.CO%3B2-V&origin=euclid
[18] --------, Dichotomies and asymptotic behaviour for linear differential systems, Trans. Amer. Math. Soc. 283 (1984), 465-484. JSTOR: · Zbl 0559.34049 · doi:10.2307/1999142 · http://links.jstor.org/sici?sici=0002-9947%28198406%29283%3A2%3C465%3ADAABFL%3E2.0.CO%3B2-X&origin=euclid
[19] Z. Nehari, Disconjugate linear differential operators, Trans. Amer. Math. Soc. 129 (1967), 500-516. JSTOR: · Zbl 0183.09101 · doi:10.2307/1994604 · http://links.jstor.org/sici?sici=0002-9947%28196712%29129%3A3%3C500%3ADLDO%3E2.0.CO%3B2-0&origin=euclid
[20] G. B. Price, Some identities in the theory of determinants, Amer. Math. Monthly 54 (1947), 75-90. JSTOR: · Zbl 0029.00210 · doi:10.2307/2304856 · http://links.jstor.org/sici?sici=0002-9890%28194702%2954%3A2%3C75%3ASIITTO%3E2.0.CO%3B2-K&origin=euclid
[21] B. Schwarz, Totally positive differential systems, Pacific J. Math. 32 (1970), 203-229. · Zbl 0193.04501 · doi:10.2140/pjm.1970.32.203
[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 · doi:10.1017/S0308210500012634
[23] --------, Some applications of Hausdorff dimension inequalities for ordinary differential equations, Proc. Roy. Soc. Edinburgh 104A (1986), 235-259. · Zbl 0622.34040 · doi:10.1017/S030821050001920X
[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