Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Global families of limit cycles of planar analytic systems


Author: L. M. Perko
Journal: Trans. Amer. Math. Soc. 322 (1990), 627-656
MSC: Primary 58F21; Secondary 34C05, 58F14
MathSciNet review: 998357
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The global behavior of any one-parameter family of limit cycles of a planar analytic system $ \dot x = f(x,\lambda )$ depending on a parameter $ \lambda \in R$ is determined. It is shown that any one-parameter family of limit cycles belongs to a maximal one-parameter family which is either open or cyclic. If the family is open, then it terminates as the parameter or the orbits become unbounded, or it terminates at a critical point or on a (compound) separatrix cycle of the system. This implies that the periods in a one-parameter family of limit cycles can become unbounded only if the orbits become unbounded or if they approach a degenerate critical point or (compound) separatrix cycle of the system. This is a more specific result for planar analytic systems than Wintner's principle of natural termination for $ n$-dimensional systems where the periods can become unbounded in strange ways. This work generalizes Duffs results for one-parameter families of limit cycles generated by a one-parameter family of rotated vector fields. In particular, it is shown that the behavior at a nonsingular, multiple limit cycle of any one-parameter family of limit cycles is exactly the same as the behavior at a multiple limit cycle of a one-parameter family of limit cycles generated by a one-parameter family of rotated vector fields.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F21, 34C05, 58F14

Retrieve articles in all journals with MSC: 58F21, 34C05, 58F14


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0998357-4
PII: S 0002-9947(1990)0998357-4
Article copyright: © Copyright 1990 American Mathematical Society