Weird quasiperiodic attractors in a piecewise linear Belykh map

Abstract We consider a two-dimensional discontinuous piecewise linear homogeneous map in two partitions, belonging to the family of Belykh maps, whose characteristic property is that the Jacobian determinant is the same at any point of the phase plane. The fixed points of the functions defining the map have disjoint stability regions. Three kinds of discontinuity sets separating these partitions are considered. As shown in our previous works, discontinuous piecewise linear homogenous maps can have weird quasiperiodic attractors (WQAs). We prove that there are parameter regions where the existence of WQAs is generic. Owing to the complex shape of a WQA, it may appear similar to a chaotic attractor. However, a WQA does not contain any cycle or other invariant subsets, being the closure of quasiperiodic trajectories. We present examples of contact bifurcations of coexisting WQAs with their basin boundaries leading to a unique WQA, showing that these bifurcations are not associated with homoclinic/heteroclinic trajectories. We discuss also contact bifurcations leading to the divergence that may be accompanied by a long transient following the former attractor. The simplicity of the map allows us to describe the mechanism of the appearance/disappearance of WQAs, always related to the unstable eigenvector of the fixed point.