A low-dimensional model of general circulation of the atmosphere is investigated. The differential equations are subject to periodic forcing, where the period is one year. A three-dimensional Poincaré mapping depends on three control parametersF, G, and epsilon, the latter being the relative amplitude of the oscillating part of the forcing. This paper provides a coherent inventory of the phenomenology of F, G, epsilon. For epsilon small, a Hopf-saddle-node bifurcation of fixed points and quasi-periodic Hopf bifurcations of invariant circles occur, persisting from the autonomous case epsilon= 0. For epsilon= 0.5, the above bifurcations have disappeared. Different types of strange attractors are found in four regions (chaotic ranges) in {F, G} and the related routes to chaos are discussed.