International Journal of Bifurcation and Chaos · 1995 · 82 citations · 0 references
Natural SciencesMathematicsComplex SystemsDynamical SystemsSystem DynamicsBifurcation TheoryChaos TheoryComplex DynamicsContinuous Dynamical SystemsNonlinear DynamicsOscillatorsBifurcationBifurcation AnalysisBifurcation Analysis In Control SystemsHomoclinic BifurcationPiecewise Linear Maps
We study the bifurcations of attracting cycles for a three-segment (bimodal) piecewise-linear continuous one-dimensional map. Exact formulas for the regions of periodicity of any rational rotation number (Arnold’s tongues) are obtained in the associated three-dimensional parameter space. It is shown that the destruction of any Arnold’s tongue is a result of a border-collision bifurcation, and is followed by the appearance of a cycle of intervals with the same rotation number, whose dynamics is determined by a skew tent map. Finally, for the interval cycle the merging bifurcation corresponds to a homoclinic bifurcation of some point cycle.