A recent method of Delamotte [Phys. Rev. Lett. 70, 3361 (1993)] for obtaining approximate analytic expressions for the loci of limit cycles in one variable is applied to coupled nonlinear first-order rate equations in several variables, the typical case for most models based on chemical kinetics. The first-order approximation works well near a bifurcation point, with higher-order terms being required the further the system is from the bifurcation point. The method complements linear stability analysis (which gives the limiting frequency of the limit cycle at the bifurcation point) by giving a simple method with which to construct an explicit formula that gives the evolution in space and time of a limit cycle near a bifurcation point.