zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay. (English) Zbl 0849.34056
The subject of the paper is the autonomous delay monotone cyclic feedback system of the form $\dot x^0 (t) = f^0 (x^0(t), x^1(t))$, $\dot x^i (t) = f^i (x^{i - 1}(t)$, $x^i (t)$, $x^{i + 1} (t))$, $1 \le i \le N - 1$, $\dot x^N(t) = f^N (x^{N - 1}(t)$, $x^N (t)$, $x^0 (t - 1))$, where the functions $f^0$, $f^1, \dots, f^N$ satisfy some monotonicity conditions with respect to their first and last arguments. The authors show that under rather mild conditions on the nonlinearities $f^0$, $f^1, \dots,f^N$ the Poincaré-Bendixson theorem holds in force: either (a) the $\omega$-limit set $\omega (x)$ of a bounded solution $x$ is a single non-constant periodic orbit; or, else, (b) all $\alpha$- and $\omega$-limit points of any solution in $\omega (x)$ are equilibrium points of the system. The most part of the paper is devoted to the proof of this result. In Section 7 the authors investigate and provide very interesting results on the behavior of the periodic solutions with an emphasis on the winding number of the curve $t \to (x^i(t), x^{i + 1} (t))$ on the plane, while in the last section they examine the connection between oscillation of a periodic solution and its instability.

34K99Functional-differential equations
34K20Stability theory of functional-differential equations
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34C25Periodic solutions of ODE
Full Text: DOI