Neimark–Sacker Bifurcations in Planar, Piecewise-Smooth, Continuous Maps

The multipliers of a fixed point of a piecewise-smooth, continuous map may change discontinuously as the fixed point crosses a discontinuity under smooth variation of parameters. We study the case when the multipliers “jump” from inside to outside the unit circle, and we assume the map is two-dimensional and piecewise-affine. The resulting dynamics is sometimes similar to the Neimark–Sacker bifurcation of a smooth map in which an attracting periodic or quasiperiodic orbit is created as the fixed point loses stability. However, the bifurcation is often much more complex, with multiple (chaotic) attractors, saddles, and repellors created or destroyed.