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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 $$\dot x^0 (t) = f^0 (x^0(t), x^1(t))$$, $$\dot x^i (t) = f^i (x^{i - 1}(t)$$, $$x^i (t)$$, $$x^{i + 1} (t))$$, $$1 \leq i \leq N - 1$$, $$\dot x^N(t) = f^N (x^{N - 1}(t)$$, $$x^N (t)$$, $$x^0 (t - 1))$$, 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.

##### MSC:
 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 34K20 Stability theory of functional-differential equations 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations
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