Asymptotic behavior of solutions to a class of systems of delay differential equations (Q2385326)

The authors investigate the asymptotic behavior of solutions of the system of delay differential equations \[ \begin{gathered} x_1'= -F(x_1(t))+ G(x_2(t- r_2)),\\ x_2'= -F(x_2(t))+ G(x_1(t- r_1)),\end{gathered} \] where \(r_1\), \(r_2\) are positive constants; \(F,G\in C(\mathbb{R})\) and either \(G(x)\geq F(x)\) or \(G(x)\leq F(x)\), \(x\in\mathbb{R}\). \(F\) is assumed nondecreasing on \(\mathbb{R}\). Under certain assumptions on \(F\), \(G\) it is shown that a bounded positive orbit has a limit set consisting of a constant vector in \(\mathbb{R}^2\).