In this paper, we construct an exact formula for the periods of solutions of the system of ordinary differential equations (1) \(\dot x=-\lambda f(y(t))\), \(\dot y=\lambda f(x(t))\). Throughout the paper, we assume that f is a function which satisfies the following conditions: \(f(x)x>0\) for \(x\neq 0\), f is odd and differentiable, \(f'(0)=1\). The formula we obtain for the periods is a function \(T(x_ 0,\lambda)\) of \(\lambda\) and \(x_ 0>0\) where \((x_ 0,0)\) is any initial data for system (1). It is known that period-4-solutions of (1) yield period-4-solutions of (2) \(\dot x=- \lambda (x(t-1))\). For these solutions, \(T(x_ 0,\lambda)\) yields a function \(\lambda\) (x,0) which shows that a branch of period-4-solutions of (2) bifurcates at \(\lambda =\pi /2\). Using \(\lambda (x_ 0)\), we show that the higher derivatives of f at 0 determine the behaviour of the branch near \(\pi\) /2.