Sushko, Iryna; Agliari, Anna; Gardini, Laura Bifurcation structure of parameter plane for a family of unimodal piecewise smooth maps: Border-collision bifurcation curves. (English) Zbl 1142.37339 Chaos Solitons Fractals 29, No. 3, 756-770 (2006). Summary: We study the structure of the 2D bifurcation diagram for a two-parameter family of piecewise smooth unimodal maps \(f\) with one break point. Analysing the parameters of the normal form for the border-collision bifurcation of an attracting \(n\)-cycle of the map \(f\), we describe the possible kinds of dynamics associated with such a bifurcation. Emergence and role of border-collision bifurcation curves in the 2D bifurcation plane are studied. Particular attention is paid also to the curves of homoclinic bifurcations giving rise to the band merging of pieces of cyclic chaotic intervals. Cited in 34 Documents MSC: 37E05 Dynamical systems involving maps of the interval 37G99 Local and nonlocal bifurcation theory for dynamical systems PDFBibTeX XMLCite \textit{I. Sushko} et al., Chaos Solitons Fractals 29, No. 3, 756--770 (2006; Zbl 1142.37339) Full Text: DOI References: [1] Banerjee, S.; Karthik, M. S.; Yuan, G.; Yorke, J. A., Bifurcations in one-dimensional piecewise smooth maps—theory and applications in switching circuits, IEEE Trans Circuits Syst I: Fund Theory Appl, 47, 3, 389-394 (2000) · Zbl 0968.37013 [2] Day, R., Irregular growth cycles, Am Econom Rev, 72, 3, 406-414 (1982) [3] Di Bernardo, M.; Feigen, M. I.; Hogan, S. J.; Homer, M. E., Local analysis of \(C\)-bifurcations in \(n\)-dimensional piecewise smooth dynamical systems, Chaos, Solitons & Fractals, 10, 11, 1881-1908 (1999) · Zbl 0967.37030 [4] Halse, C.; Homer, M.; di Bernardo, M., C-bifurcations and period-adding in one-dimensional piecewise-smooth maps, Chaos, Solitons & Fractals, 18, 953-976 (2003) · Zbl 1069.37033 [5] Hao, B. L., Elementary symbolic dynamics and chaos in dissipative systems (1989), World Scientific: World Scientific Singapore · Zbl 0724.58001 [6] Ito, S.; Tanaka, S.; Nakada, H., On unimodal transformations and chaos II, Tokyo J Math, 2, 241-259 (1979) · Zbl 0461.28017 [7] Kowalczyk P, di Bernardo M, Champneys AR, Hogan SJ, Homer M, Kuznetsov YuA, et al. Two-parameter nonsmooth bifurcations of limit cycles: classification and open problems. Int J Bifurcat Chaos, submitted for publication.; Kowalczyk P, di Bernardo M, Champneys AR, Hogan SJ, Homer M, Kuznetsov YuA, et al. Two-parameter nonsmooth bifurcations of limit cycles: classification and open problems. Int J Bifurcat Chaos, submitted for publication. [8] Maistrenko, Yu. L.; Maistrenko, V. L.; Chua, L. O., Cycles of chaotic intervals in a time-delayed Chua’s circuit, Int J Bifurcat Chaos, 3, 6, 1557-1572 (1993) · Zbl 0890.58052 [9] Maistrenko, Yu. L.; Maistrenko, V. L.; Vikul, S. I., On period-adding sequences of attracting cycles in piecewise linear maps, Chaos, Solitons & Fractals, 9, 1, 67-75 (1998) · Zbl 0934.37034 [10] Metropolis, N.; Stein, M. L.; Stein, P. R., On finite limit sets for transformations on the unit interval, J Comb Theory, 15, 25-44 (1973) · Zbl 0259.26003 [11] Mira, C., Chaotic dynamics (1987), World Scientific: World Scientific Singapore [12] Misiurewicz, M.; Kawczynski, A. L., Periodic orbits for interval maps with sharp cusps, Physica D, 52, 191-203 (1991) · Zbl 0741.58013 [13] Nusse, H. E.; Yorke, J. A., Border-collision bifurcations including period two to period three for piecewise smooth systems, Physica D, 57, 39-57 (1992) · Zbl 0760.58031 [14] Nusse, H. E.; Yorke, J. A., Border-collision bifurcations for piecewise smooth one-dimensional maps, Int J Bifurcat Chaos, 5, 1, 189-207 (1995) · Zbl 0885.58060 [15] Sushko, I.; Agliari, A.; Gardini, L., Bistability and border-collision bifurcations for a family of unimodal piecewise smooth maps, Discrete Contin Dynam Syst Ser B, 5, 3, 881-897 (2005) · Zbl 1087.37029 [16] Zhusubaliyev, Z. T.; Mosekilde, E., Bifurcations and chaos in piecewise-smooth dynamical systems (2003), World Scientific: World Scientific Singapore · Zbl 1047.34048 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.