## The Poincaré-Bendixson theorem for monotone cyclic feedback systems.(English)Zbl 0712.34060

The authors consider systems of the form (*) $$\dot x^ i=f^ i(x^ i,x^{i-1})$$, $$i=1,2,...,n$$, where we agree to interpret $$x^ 0$$ as $$x^ n$$. They assume that the nonlinearity $$f=(f^ 1,f^ 2,...,f^ n)$$ is defined on a nonempty open set $$U\subset R^ n$$ with the property that each coordinate projection $$U^ i\subset {\mathbb{R}}^ 2$$ of U onto the $$(x^ i,x^{i-1})$$-plane is convex and that $$f^ i\in C^ 1(U^ i).They$$ assume also for some $$\delta^ i\in \{1,+1\}$$, that (**): $$\delta^ i\partial f^ j(x^ i,x^{i-1})/\partial x^{i-1}>0$$ for all $$(x^ i,x^{i-1})\in U^ i$$ and $$i=1,2,...,n$$. Such a system, of the form (*) satisfying (**), is called a monotone cyclic feedback system. The main result of this paper is that the Poincaré-Bendixson theorem holds for monotone cyclic feedback systems. The organization of this paper is as follows. In § 1 a principal tool, an integer valued Lyapunov function N is developed. § 2 is concerned with the Floquet theory of linear monotone cyclic feedback systems. § 3 is devoted to the proof of the main result. Finally, in § 4 various applications are treated.
Reviewer: Chungyou He

### MSC:

 34C25 Periodic solutions to ordinary differential equations 93C15 Control/observation systems governed by ordinary differential equations
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### References:

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