Sabatini, M. Bifurcation of periodic solutions from inversion of stability of periodic O.D.E.’s. (English) Zbl 0794.34034 Rend. Semin. Mat. Univ. Padova 89, 1-9 (1993). A parameterized family of differential equations \(x'= f(\lambda,t,x)\), \(f\) \(T\)-periodic in \(t\), where \(\lambda\in [0,\lambda_ 0)\) and \(x\in \mathbb{R}^ n\) is considered. The main result states that if \(n\) is odd, \(\{u_ \lambda\}\) is a family of \(T\)-periodic solutions continuous with respect to \(\lambda\), \(u_ 0\) is asymptotically stable and \(u_ \lambda\) is negatively asymptotically stable for \(\lambda>0\), then \(\lambda=0\) is a bifurcation parameter for \(\{u_ \lambda\}\). The result is a consequence of the Lefschetz Fixed Point theorem applied to the corresponding Poincaré operator restricted to the set bounded by two concentric spheres. Reviewer: R.Srzednicki (Buffalo) MSC: 34C23 Bifurcation theory for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations 34D20 Stability of solutions to ordinary differential equations Keywords:asymptotical stability; periodic solutions; bifurcation parameter; Lefschetz Fixed Point theorem; Poincaré operator PDFBibTeX XMLCite \textit{M. Sabatini}, Rend. Semin. Mat. Univ. Padova 89, 1--9 (1993; Zbl 0794.34034) Full Text: Numdam EuDML References: [1] N.P. Bathia - G.P. Szegö , Stability theory of dynamical systems , Die Grund. der Math. Wiss. in Einz. , vol. 161 , Springer-Verlag , Berlin ( 1970 ). MR 289890 · Zbl 0213.10904 [2] S.R. Bernfeld - L. Salvadori - F. Visentin , Discrete dynamical systems and bifurcation for periodic differential equation , Nonlinear Anal., Theory, Methods Appl. , 12 - 9 ( 1988 ), pp. 881 - 893 . MR 960633 | Zbl 0653.34031 · Zbl 0653.34031 · doi:10.1016/0362-546X(88)90072-7 [3] S.N. Chow - J. K. HALE, Methods of Bifurcation Theory , Springer , New York ( 1982 ). MR 660633 | Zbl 0487.47039 · Zbl 0487.47039 [4] W. Hahn , Stability of Motion , Die Grund. der Math. Wiss. in Einz. , vol. 138 , Springer-Verlag , Berlin ( 1967 ). MR 223668 | Zbl 0189.38503 · Zbl 0189.38503 [5] J. Hale , Ordinary Differential Equations , Pure Appl. Math. , vol. XXI , Wiley-Interscience , New York ( 1980 ). MR 587488 | Zbl 0186.40901 · Zbl 0186.40901 [6] F. Marchetti - P. Negrini - L. Salvadori - M. Scalia , Liapunov direct method in approaching bifurcation problems , Ann. Mat. Pura Appl. ( IV ), 108 ( 1976 ), pp. 211 - 226 . MR 445076 | Zbl 0332.34047 · Zbl 0332.34047 · doi:10.1007/BF02413955 [7] L. Salvadori , Bifurcation and stability problems for periodic differential systems , Proc. Conf. on Non Linear Oscillations of Conservative Systems ( 1985 ), pp. 305 - 317 (A. AMBROSETTI, ed.), Pitagora , Bologna . · Zbl 0577.34044 [8] G. Sansone - R. Conti , Non-linear Differential Equations , MacMillan Company , New York ( 1964 ). MR 177153 | Zbl 0128.08403 · Zbl 0128.08403 [9] E.H. Spanier , Algebraic Topology , McGraw-Hill , Bombay ( 1966 ). MR 210112 | Zbl 0145.43303 · Zbl 0145.43303 [10] T. Yoshizawa , Stability Theory by Liapunov’s Second Method , The Mathematical Society of Japan , Tokio ( 1966 ). MR 208086 | Zbl 0144.10802 · Zbl 0144.10802 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.