ArticleOriginal scientific text
Title
Generic saddle-node bifurcation for cascade second order ODEs on manifolds
Authors 1
Affiliations
- Department of Mathematical Analysis, Faculty of Mathematics and Physics, Comenius University, Mlynská dolina, 842 15 Bratislava, Slovakia
Abstract
Cascade second order ODEs on manifolds are defined. These objects are locally represented by coupled second order ODEs such that any solution of one of them can represent an external force for the other one. A generic saddle-node bifurcation theorem for 1-parameter families of cascade second order ODEs is proved.
Keywords
cascade, ODE, critical element, transversal, bifurcation
Bibliography
- R. Abraham and J. Robbin, Transversal Mappings and Flows, W. A. Benjamin, New York, 1967.
- M. Medveď, Oscillatory properties of some classes of nonlinear differential equations, Math. Bohemica 117 (1992), 95-107.
- M. Medveď, Generic bifurcations of second order ordinary diferential equations on differentiable manifolds, Math. Slovaca 27 (1977), 9-24.
- M. Medveď, Generic bifurcations of second order ordinary differential equations near closed orbits, J. Differential Equations 36 (1980), 98-107.
- M. Medveď, On generic bifurcations of second order ordinary differential equations near closed orbits, Astérisque 50 (1977), 23-29.
- M. Medveď, Fundamentals of Dynamical Systems and Bifurcation Theory, Adam Hilger, Bristol, 1992.
- M. Medveď, A class of vector fields on manifolds containing second order ODEs, Hiroshima Math. J. 26 (1996), 127-149.
- P. Seibert and R. Suarez, Global stabilization of nonlinear cascade systems, Systems Control Lett. 14 (1990), 347-352.
- S. Shahshahani, Second order ordinary differential equations on differentiable manifolds, in: Global Analysis, Proc. Sympos. Pure Math. 14, Amer. Math. Soc., 1970, 265-272.
Pages:
211-225
Main language of publication
English
Received
1997-01-23
Accepted
1997-07-17
Published
1998
Exact and natural sciences