Dynamics analysis of a class of planar systems with time-varying delays (Q867972)

The authors consider a planar system of the following form: \[ x'_1(t)=-a_1(t)x_1(t)+b_1(t)f_1(x_1(t-\tau_{11}(t)),x_2(t-\tau_{12}(t)))+I_1(t), \] \[ x'_2(t)=-a_2(t)x_2(t)+b_2(t)f_2(x_1(t-\tau_{21}(t)),x_2(t-\tau_{22}(t)))+I_2(t), \] where \(a_i\in C(R, (0,\infty)),\; b_i,\; I_i\in C(R,R),\;i=1,2,\) are periodic with a common period \(\omega>0,\; f_i\in C(R^2, R),\) the functions \(\tau_{ij}\in C(R,[0,\infty)),\;i,j=1,2,\) are \(\omega\)-periodic, smoothness, monotonicity and boundedness of the activation functions are not assumed. Using the coincidence degree theory, they obtain the existence of an \(\omega\)-periodic solution to the above-mentioned system. At the end of this paper some sufficient conditions for the global exponential stability of a periodic solution and an example illustrating the main results are also presented.