Abstract
Two-dimensional autonomous systems may have solution curves which form separatrix polygons in the phase plane. These are polygons the corner points of which are saddle points and the sides of which are separatrices connecting these saddle points. They are structurally unstable, and in this paper we will study the change in the phase portrait due to arbitrary small changes in the right hand sides of these systems. In particular, attention will be given to the generation of limit cycles from these polygons. The number of limit cycles generated by a separatrix polygon is seen to be related to the eigenvalues of the locally linearized system in the saddle points. For separatrix polygons with two saddle points, criteria, involving these eigenvalues are given when exactly one or exactly two limit cycles can be generated. For separatrix polygons with three or more saddle points similar criteria are given to ensure the generation of at least one, two, or more (till n for a n sided polygon) limit cycles.
Keywords
- Saddle Point
- Phase Portrait
- Limit Continuum
- Stable Limit Cycle
- Closed Path
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References
Andronov, A.A., Gordon, J.J., Leontovich, E.A. and Maier, A.G.; Qualitative Theory of Second-Order Dynamic Systems, Israel Program for Scientific Translation, Jerusalem, 1973.
Andronov, A.A., Gordon, J.J., Leontovich, E.A. and Maier, A.G.; Theory of Bifurcations of Dynamic Systems on a Plane, Israel Program for Scientific Translation, Jerusalem, 1971.
Comtet, L.; Advanced Combinatorics; the art of finite and infinite expansions; revised and enlarged edition, Reidel, Dordrecht, 1974.
Dulac, H.; Sur les cycles limites, Bull. Soc. Math. de France, Vol. 51, pp. 45–188, 1923.
Jablonski; Théorie des permutations et des arrangements completes, Journal de Liouville, 8, pp. 331–349, 1892.
Leontovich, E.A.; On the generation of limit cycles from separatrices, Dokl. Akad. Nank. U.S.S.R., Vol. 78, no. 4, pp. 641–644, 1951.
Reyn, J.W.; A stability criterion for separatrix polygons in the phase plane, Nieuw Archief voor Wiskunde (3), XXVII, pp. 238–254, 1979.
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© 1980 Springer-Verlag
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Reyn, J.W. (1980). Generation of limit cycles from separatrix polygons in the phase plane. In: Martini, R. (eds) Geometrical Approaches to Differential Equations. Lecture Notes in Mathematics, vol 810. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0089983
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DOI: https://doi.org/10.1007/BFb0089983
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-10018-8
Online ISBN: 978-3-540-38166-2
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