A first order system of ordinary differential equations containing two real parameters may have a simple bifurcation of steady states and a Hopf bifurcation of periodic solutions existing at nearby points of parameter space. The nonlinear interactions between these two bifurcating solutions are shown to give rise to secondary Hopf bifurcations in the generic case, and to doubly-periodic solutions on a torus for a significant class of systems. The paper presents a classification of the possible mode interactions, asymptotic formulae for the bifurcating solutions, and iterative algorithms for their numerical computation. The results are applied to examples drawn from the literature of nonlinear hydrodynamics and of mathematical ecology.