Cones of rank 2 and the Poincaré-Bendixson property for a new class of monotone systems
To state the main result of the paper about monotone systems, we introduce some notation. A set \(C\subset\mathbb R^n\) is a cone of rank \(k\) if it is closed; \(x\in C\), \(\alpha\in\mathbb R\) imply that \(\alpha x\in C\) and \(\max\{\dim (W): C\supset W\) linear subspace\(\}=k\). The closure of the set \(\mathbb R^n\setminus C\) is also a cone, called the complementary cone and denoted by \(C^c\). A cone \(C\) is \(k\)-solid if there is a linear subspace \(W\) of dimension \(k\) such that \(W\setminus\{0\}\subset \text{Int}(C).\) Consider a \(C^1\) autonomous differential system \(\dot x=F(x)\) and suppose that it induces a semiflow \(\varphi(t,x)\) defined for all \(t\geq0.\) Given two arbitrary points \(p,q\in\mathbb R^n\) consider the two matrices \[ A^{pq}(t)= \int_0^1 DF\big(s\varphi(t,p)+(1-s)\varphi(t,q)\big)\,ds \] and \(U^{pq}(t)\) given by the solution of the Cauchy problem \(\dot U=A^{pq}(t)U,\) \(U(0)=Id.\) It is said that the differential equation \(\dot x=F(x)\) is \(C\)-cooperative if for every \(p,q\in \mathbb R^n\) it holds that for \(t>0\), the matrix \(U^{pq}(t)\) is strongly positive, that is \(U^{pq}(t)\big(C\setminus\{0\}\big)\subset \text{Int}(C).\) The author proves the following result of Poincaré-Bendixson type: Let \(C\subset \mathbb R^n\) be a cone of rank 2 such that \(C\) is 2-solid and \(C^c\) is \((n-2)\)-solid. Let us suppose that the \(C^1\) differential system \(\dot x=F(x)\) is \(C\)-cooperative ant let \(x(t)\) be a solution having a compact omega-limit set \(\Omega\) not containing equilibrium points and such that \(\dot x(t_0)\in C\) for some real value \(t_0.\) Then \(\Omega\) is a closed orbit.
scientific article; zbMATH DE number 5530627
scientific article