Bifurcation Patterns in Homogeneous Area-Preserving Piecewise-Linear Maps

The dynamical behavior of a family of planar continuous piecewise linear maps with two zones is analyzed. Assuming homogeneity and preservation of areas we obtain a canonical form with only two parameters: the traces of the two matrices defining the map. It is shown the existence of sausage-like structures made by lobes linked at the nodes of a nonuniform grid in the parameter plane. In each one of these structures, called resonance regions, the rotation number of the associated circle map is a given rational number. The boundary of the lobes and a significant inner partition line are studied with the help of some Fibonacci polynomials.