Stability of large cycles in a nonsmooth problem with Hopf bifurcation at infinity. (English) Zbl 0963.34034

From the introduction: The results deal with the stability of large cycles where Hopf bifurcation at infinity occurs. Here, the authors develop a new method to study both existence and stability of large cycles for systems with nonsmooth nonlinearities, asymptotically homogeneous at infinity. The method combines the parameter functionalization technique and methods of monotone concave operators.


34D05 Asymptotic properties of solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
Full Text: DOI


[1] Glober, J., Hopf bifurcation at infinity, Nonlinear anal., 11, 1393-1398, (1989) · Zbl 0705.34042
[2] B. Hassard, N. Kazarinoff, Y.-H. Wan, Theory and Applications of Hopf Bifurcations, Cambridge University Press, London, 1981. · Zbl 0474.34002
[3] Xiangian, He, Hopf bifurcation at infinity with discontinuous nonlinearities, J. austral. math. soc. B., 33, 133-148, (1991) · Zbl 0806.34027
[4] Kozyakin, V.S.; Krasnosel’skii, M.A., The method of parameter functionalization in the Hopf bifurcation problem, 11, 149-161, (1987) · Zbl 0661.34036
[5] M.A. Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
[6] Krasnosel’skii, A.M.; Krasnosel’skii, M.A., Large-amplitude cycles in autonomous systems with hysteresis, Soviet math. dokl., 32, 1, 14-17, (1985) · Zbl 0602.34024
[7] Krasnosel’skii, M.A.; Kuznetsov, N.A.; Yumagulov, M.G., Localization and construction of cycles in Hopf’s bifurcation at infinity, Dokl. math., 52, 2, 223-226, (1995) · Zbl 0880.34038
[8] Krasnosel’skii, M.A.; Kuznetsov, N.A.; Yumagulov, M.G., Conditions of cycle stability for the Hopf bifurcation at infinity, Automat. remote control, 58, 1, 43-48, (1997) · Zbl 0920.34040
[9] M.A. Krasnosel’skii, P.P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin, 1984.
[10] J. Marsden, M. McCracken, Hopf Bifurcation and its Applications, Springer, New York, 1982. · Zbl 0346.58007
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.