Abundance of Weird Quasiperiodic Attractors in Piecewise Linear Discontinuous Maps

In this work, we consider a class of n-dimensional, n >= 2, piecewise linear discontinuous maps that can exhibit a new type of attractor, called a weird quasiperiodic attractor. While the dynamics associated with these attractors may appear chaotic, we prove that chaotic attractors cannot occur. The considered class of n-dimensional maps allows for any finite number of partitions, separated by various types of discontinuity sets. The key characteristic, beyond discontinuity, is that all functions defining the map have the same real fixed point. These maps cannot have hyperbolic cycles other than the fixed point itself. We consider the two-dimensional case in detail. We prove that in nongeneric cases, the restriction, or the first return, of the map to a segment of straight line issuing from the fixed point is reducible to a piecewise linear circle map. The generic attractor, different from the fixed point, is a weird quasiperiodic attractor, which may coexist with other attractors or attracting sets. We illustrate the existence of these attractors through numerous examples, using functions with different types of Jacobian matrices, as well as with different types of discontinuity sets. An application to a financial market modeling shows the role of regulator that maps in our class can have, leading to endogenous nonregular dynamics. In some cases, we describe possible mechanisms leading to the appearance of these attractors. We also give examples in the three-dimensional space. Several properties of this new type of attractor remain open for further investigation.