On the construction of a periodic solution of a second-order delay system (Q5930616)

The author deals with a system of delay differential equations arising in a problem of macroeconomic analysis \[ \dot{x}(t)=(a_0-a_1x(t-h)-a_2y(t-h))x(t),\quad \dot{y}(t)=(b_0-b_1x(t-h)-b_2y(t-h))y(t), \] where \(x(t)\) and \(y(t)\) characterize two industries competing in a common market, \(a_0,a_1,a_2,b_0,b_1,\) and \(b_2\) are positive constants, and \(h\) is the time lag between the current state of an industry and the state determining the reproduction coefficients of the industries. The main result of the paper establishes the existence of a periodic solution in a neighborhood of the nontrivial critical point \(P_4\) for any finite delay. An iterative process for the construction of such a periodic solution is proposed.