Dynamics of a Family of Continuous Piecewise Linear Planar Maps

For the family of piecewise linear maps Fa,b(x, y) = (|x| - y + a, x - |y| + b), the dynamics strongly depend on the parameters a and b. When a ≥ 0, all orbits are eventually periodic, while for a < 0, the dynamics is concentrated on one-dimensional invariant graphs that capture the final dynamics of the map.