Wang, Kaihua; Fu, Xinchu Discrete Conley index and bifurcation points. (English) Zbl 1240.37015 J. Shanghai Univ. 14, No. 6, 400-404 (2010). Summary: A sufficient condition for the existence of bifurcation points for discrete dynamical systems is presented. The relation between two families of systems is further discussed, and a sufficient condition for determining whether they may have similar bifurcation points is given. MSC: 37B30 Index theory for dynamical systems, Morse-Conley indices 37G99 Local and nonlocal bifurcation theory for dynamical systems Keywords:discrete dynamical system; prime isolated invariant set; extreme maximal isolated invariant set; Conley index; bifurcation point PDFBibTeX XMLCite \textit{K. Wang} and \textit{X. Fu}, J. Shanghai Univ. 14, No. 6, 400--404 (2010; Zbl 1240.37015) Full Text: DOI arXiv References: [1] Conley C. Isolated invariant sets and the Morse index [M]. Rhode Island: American Mathematical Society, 1978: 1–81. · Zbl 0397.34056 [2] Chow S N, Hale J K. Methods of bifurcation theory [M]. Berlin: Springer-Verlag, 1982: 97–150. · Zbl 0487.47039 [3] Li C P, Yang Z H, Chen G R. On bifurcation from steady-state solutions to rotating waves in the Kuramoto-Sivashinsky equation [J]. Journal of Shanghai University (English Edition), 2005, 9(4): 286–291. · Zbl 1084.65094 · doi:10.1007/s11741-005-0038-6 [4] Peng M J, Cheng Y M. Some problems in nonlinear dynamic instability and bifurcation theory for engineering structures [J]. Journal of Shanghai University (English Edition), 2005, 9(1): 29–34. · Zbl 1082.74022 · doi:10.1007/s11741-005-0100-4 [5] Fu X C, Xu K H. The Conley index and bifurcation points [J]. Nonlinear Analysis TMA, 1992, 19(12): 1137–1142. · Zbl 0771.58034 · doi:10.1016/0362-546X(92)90187-J [6] Fu X C. The intension subdivision and existence conditions of bifurcation points [J]. Journal of Nonline Dynamics in Science and Technology, 1995, 2(2): 99–105. [7] Mrozek M. Leary functor and the cohomological Conley index for discrete dynamical systems [J]. Transactions of the American Mathematical Society, 1990, 318(1): 149–178. · Zbl 0686.58034 · doi:10.1090/S0002-9947-1990-0968888-1 [8] Mrozek M, Rybakowski K P. A cohomological Conley index for maps on metric spaces [J]. Journal of Differential Equations, 1991, 90(1): 143–171. · Zbl 0721.58040 · doi:10.1016/0022-0396(91)90165-6 [9] Mrozek M. Shape index and other indices of Conley type for local maps on locally compact hausdorff spaces [J]. Fundamenta Mathematicae, 1994, 145(1): 15–37. · Zbl 0870.54043 [10] Robbin J W, Salamon D. Dynamical systems, shape theory and the Conley index [J]. Ergodic Theory and Dynamical Systems, 1988(8): 375–393. · Zbl 0682.58040 · doi:10.1017/S0143385700009494 [11] Kaczynski T, Mrozek M. Connected simple systems and the Conley functor [J]. Topological Methods in Nonlinear Analysis, 1997, 10(1): 183–193. · Zbl 0909.54033 [12] Szymczak A. The Conley index for discrete semidynamical systems [J]. Topology and its Applications, 1995, 66(3): 215–240. · Zbl 0840.34043 · doi:10.1016/0166-8641(95)0003J-S [13] Kuznetsov Y A. Elements of applied bifurcation theory [M]. 3rd ed. Berlin: Springer-Verlag, 2004: 39–41. · Zbl 1082.37002 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.