Using a photospheric magnetogram of the solar activity complex HR 16862, 16863, 16864 at the end of May 1980 as boundary data a sequence of three-dimensional force-free magnetic fields with spatially constant α (α defined by the equation ▽ × B = αB) is calculated, including numerical field-line tracing. The field is assumed not to be frozen-in and ¦α¦, which is a measure of the magnetic free energy, is assumed to increase with time due to some dynamo mechanism. Variation of α, starting from α = 0, produces a catastrophe-like change of the topology of a field-line system corresponding to an arch filament system connecting the leading spot of HR 16863 with the trailing spot of HR 16862. This topological change is interpreted as causing a series of large homologous flares with synchronous flaring in the two spots. The catastrophe-like behaviour of the field topology is attributed to the nonlinear field-line equations.