Simultaneous border-collision and period-doubling bifurcations.
We unfold the codimension-two simultaneous occurrence of a border-collision bifurcation and a period-doubling bifurcation for a general piecewise-smooth, continuous map. We find that with sufficient nondegeneracy conditions, a locus of period-doubling bifurcations emanates nontangentially from a locus of border-collision bifurcations. The corresponding period-doubled solution undergoes a border-collision bifurcation along a curve emanating from the codimension-two point and tangent to the period-doubling locus here. In the case that the map is one-dimensional local dynamics is completely classified; in particular, we give conditions that ensure chaos.
Simpson
D J W
DJ
Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309-0526, USA.
Meiss
J D
JD
D000465:Algorithms; D003198:Computer Simulation; D015233:Models, Statistical; D017711:Nonlinear Dynamics; D009991:Oscillometry / Q000379:methods*