We study the effect of variable external forcing on a dynamical system proposed by Lorenz as perhaps the simplest possible 'general circulation' model of the atmosphere. When a strong annual cycle is included, numerical exploration reveals the existence of a variety of multiyear climate states, which fall into two basic types. In the space of external forcing parameters, the different kinds of climate state are interleaved in an intricate pattern at scales ≤0.01%. Since this is below the ~0.1% level of observed solar cycle irradiance variability, then the model climate state can be modulated by the solar variability. If the solar cycle is accompanied by a steady drift in forcing, it can produce periodic modulation that appears, disappears, and even reverses its phase. A parametric drift by itself produces intervals of steady, but sometimes differing, climates punctuated by intermittent bursts of variability. Different forcing parameters for the Northern and Southern Hemispheres produce different responses to variable forcing. We test the stability of the results under the addition of noise and changes of system parameters. ¿ 2000 American Geophysical Union