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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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