Generalized Hopf Bifurcation for Planar Vector Fields via the Inverse Integrating Factor
- 139 Downloads
In this paper we study the maximum number of limit cycles that can bifurcate from a focus singular point p 0 of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider p 0 being a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of p 0 the differential system can always be brought, by means of a change to (generalized) polar coordinates (r, θ), to an equation over a cylinder in which the singular point p 0 corresponds to a limit cycle γ 0. This equation over the cylinder always has an inverse integrating factor which is smooth and non-flat in r in a neighborhood of γ 0. We define the notion of vanishing multiplicity of the inverse integrating factor over γ 0. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the singular point p 0 in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of singular points, namely for the three types of focus considered in the previous paragraph and for any isolated singular point with at least one non-zero eigenvalue.
KeywordsInverse integrating factor Generalized Hopf bifurcation Poincaré map Limit cycle Nilpotent focus
Mathematics Subject Classification (2000)37G15 37G10 34C07
Unable to display preview. Download preview PDF.
- 2.Andreev A.: Investigation on the behaviour of the integral curves of a system of two differential equations in the neighborhood of a singular point. Transl. Am. Math. Soc. 8, 187–207 (1958)Google Scholar
- 4.Andronov A.A. et al.: Theory of Bifurcations of Dynamic Systems on a Plane. Wiley, New York (1973)Google Scholar
- 5.Arnol’d, V.I.: Chapitres supplémentaires de la théorie des équations différentielles ordinaires. Reprint of the 1980 edition. “Mir”, Moscow (1984)Google Scholar
- 11.Dumortier, F., Roussarie, R.: Tracking limit cycles escaping from rescaling domains. In: Dynamical Systems and Related Topics (Nagoya, 1990). Advanced Series in Dynamical Systems, vol. 9, pp. 80–99. World Scientific Publishing, River Edge (1991)Google Scholar
- 14.Fulton W.: Algebraic Topology. A First Course. Graduate Texts in Mathematics, vol. 153. Springer, New York (1995)Google Scholar
- 23.Hale J.K., Koçak H.: Dynamics and Bifurcations. Texts in Applied Mathematics, vol. 3. Springer, New York (1991)Google Scholar
- 25.Lyapunov A.M.: Stability of Motion. Mathematics in Science and Engineering, vol. 30. Academic Press, New York (1966)Google Scholar
- 29.Narasimhan, R.: Analysis on Real and Complex Manifolds. Reprint of the 1973 edition. North-Holland Mathematical Library, vol. 35, Amsterdam (1985)Google Scholar