The following system is considered: x = k 1+αx 2 n −b 1x 1 , x =g j−1−b jx j , j=2,…, n. The equations describe a biochemical reaction system of n constituents. The variable x j is the concentration of chemical j and thus must be positive. The constants α, k, b j and g j are all positive reaction constants. The existence of periodic solutions in an arbitrarily large system is considered and resolved using dual input describing function theory. Let S = { k,α,g 1,…, g n−1 , b 1,…, b n }. A parameter set S is permissible if each element is positive. First, it is shown for n ⩽ 7 that it is impossible to construct a permissible S whose corresponding differential equation possesses a limit cycle such that x j ( t) ⩾ 0 for all t ⩾ 0. Second, it is shown that for n ⩾ 8 it is possible to construct a permissible S such that the differential equation has a stable periodic solution giving x j ( t) ⩾ 0 for all t ⩾ 0. It is possible to make an explicit statement about the S which makes a stable limit cycle possible. Define d 1 = g 1 g 2⋯ g n−1 k, d 2 = α, c 0 = b 1⋯ b n , A = d 1 d 2 / 2 c 0 and G(iω) ▪. A stable limit cycle exists if and only if the following inequality is satisfied: 1< G(iω c) G(0) 2− 2 A 2 + 4 A 4 + 8 27A 6 1 2 1 3 − 2 A 2 − 4 A 4 + 8 27A 6 1 2 1 3 ,