  We study two-dimensional, two-piece, piecewise-linear maps having two saddle
fixed points. Such maps reduce to a four-parameter family and are well known to
have a chaotic attractor throughout open regions of parameter space. The
purpose of this paper is to determine where and how this attractor undergoes
bifurcations. We explore the bifurcation structure numerically by using
Eckstein's greatest common divisor algorithm to estimate from sample orbits the
number of connected components in the attractor. Where the map is
orientation-preserving the numerical results agree with formal results obtained
previously through renormalisation. Where the map is orientation-reversing or
non-invertible the same renormalisation scheme appears to generate the
bifurcation boundaries, but here we need to account for the possibility of some
stable low-period solutions. Also the attractor can be destroyed in novel
heteroclinic bifurcations (boundary crises) that do not correspond to simple
algebraic constraints on the parameters. Overall the results reveal a broadly
similar component-doubling bifurcation structure in the orientation-reversing
and non-invertible settings, but with some additional complexities.
