Nusse, Helena E.; Yorke, James A. Border-collision bifurcations including “period two to period three” for piecewise smooth systems. (English) Zbl 0760.58031 Physica D 57, No. 1-2, 39-57 (1992). The authors study continuous maps of the plane to itself that are differentiable on either side of a smooth curve \(\Gamma\). In a one- parameter family of such maps, a fixed point may cross \(\Gamma\) and change type drastically. If the orbit index [J. Mallet-Paret and the second author, J. Differ. Equations 43, 419-450 (1982; Zbl 0487.34038)] of the fixed point changes, then there must be at least one additional periodic orbit nearby. The authors concentrate on the case in which a “flip saddle” (eigenvalues \(\lambda < -1 < v < 1\)) changes to a repeller with complex eigenvalues. They present numerical results illustrating a variety of interesting phenomena in this case. For example, a period \(p\) attractor near the flip saddle can shrink to the fixed point and reappear as a period \(q\) attractor near the repeller, for a wide variety of \(p\) and \(q\). Similarly, a period \(p\) attractor or a \(p\)-piece chaotic attractor can shrink and reappear as a \(q\)-piece chaotic attractor. One example indicates a bifurcation of chaotic invariant sets of saddle type. Most of these phenomena are illustrated for continuous maps that are linear on either side of the \(y\)-axis. The authors also give a proof that the bifurcation of a period 2 to a period 3 attractor occurs in certain families of piecewise linear maps. Reviewer: S.Schecter (Raleigh) Cited in 2 ReviewsCited in 181 Documents MSC: 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 37G99 Local and nonlocal bifurcation theory for dynamical systems Keywords:periodic orbit; attractor; repeller; bifurcation; continuous maps Citations:Zbl 0487.34038 PDFBibTeX XMLCite \textit{H. E. Nusse} and \textit{J. A. Yorke}, Physica D 57, No. 1--2, 39--57 (1992; Zbl 0760.58031) Full Text: DOI References: [1] Alligood, K. T.; Yorke, E. D.; Yorke, J. A., Why period-doubling cascades occur: period orbit creation followed by stability shedding, Physica D, 28, 197-205 (1987) · Zbl 0623.58048 [2] Guckenheimer, J.; Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, (Applied Mathematical Sciences, Vol. 42 (1983), Springer: Springer Berlin) · Zbl 0515.34001 [3] Grebogi, C.; Ott, E.; Pelikan, S.; Yorke, J. A., Strange attractors that are not chaotic, Physica D, 11, 261-268 (1984) · Zbl 0588.58036 [4] Hommes, C. H.; Nusse, H. E., “Period three to period two” bifurcation for piecewise linear models, J. Economics, 54, 157-169 (1991) · Zbl 0754.90011 [5] Hommes, C. H.; Nusse, H. E.; Simonovits, A., Hicksian cycles and chaos in a socialist economy, (Research memorandum 382 (1990), Institute of Economic Research, University of Groningen) · Zbl 0875.90106 [6] Keller, H. B., Numerical methods in bifurcation problems (1987), Springer: Springer Berlin · Zbl 0656.65063 [7] Lozi, R., Un attracteur étrange? du type attracteur de Hénon, J. Phys. (Paris), 5, C5, 9-10 (1978) [8] Mallet-Paret, J.; Yorke, J. A., Snakes: oriented families of periodic orbits, their sources, sinks and continuation, J. Diff. Eq., 43, 419-450 (1982) · Zbl 0487.34038 [9] Nusse, H. E.; Yorke, J. A., A procedure for finding numerical trajectories on chaotic saddles, Physica D, 36, 137-156 (1989) · Zbl 0728.58027 [10] Ruelle, D., Elements of Differentiable Dynamics and Bifurcation Theory (1989), Academic: Academic London · Zbl 0684.58001 [11] Seydel, R., From Equilibrium to Chaos. Practical Bifurcation and Stability Analysis (1988), Elsevier: Elsevier Amsterdam · Zbl 0652.34059 [12] Yorke, J. A., DYNAMICS: An interactive program for IBM PC clones (1990) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.