Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1990-0998357-4
MathSciNet review: 998357
Full-text PDF

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?)

  • [1] G. F. D. Duff, Limit cycles and rotated vector fields, Ann. of Math. 67 (1953), 15-31. MR 0053301 (14:751c)
  • [2] A. Wintner, Beweis des E. Stromgrenschen dynamischen Abschlusprinzips der periodischen Bahngruppen im restringierten Dreikorperproblem, Math. Z. 34 (1931), 321-349.
  • [3] L. M. Perko, On the accumulation of limit cycles, Proc. Amer. Math. Soc. 99 (1987), 515-526. MR 875391 (88b:34040)
  • [4] J. P. Francoise and C. C. Pugh, Keeping track of limit cycles, J. Differential Equations 65 (1986), 139-157. MR 861513 (88a:58162)
  • [5] C. Sparrow, The Lorentz equations: bifurcations, chaos, and strange attractors, Appl. Math. Sci., vol. 41, Springer-Verlag, New York, 1982. MR 681294 (84b:58072)
  • [6] A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Theory of bifurcations of dynamical systems on a plane, Israel Program for Scientific Translations, Jerusalem, 1971. MR 0344606 (49:9345)
  • [7] L. M. Perko, Rotated vector fields and the global behavior of limit cycles for a class of quadratic systems in the plane, J. Differential Equations 18 (1975), 63-86. MR 0374552 (51:10752)
  • [8] J. Mallet-Paret and J. A. Yorke, Snakes: oriented families of periodic orbits, their sources, sinks and continuation, J. Differential Equations 43 (1982), 419-450. MR 649847 (84a:58071a)
  • [9] K. T. Alligood and J. A. Yorke, Families of periodic orbits: virtual periods and global continuity, J. Differential Equations 55 (1984), 59-71. MR 759827 (86a:58082)
  • [10] S. N. Chow, J. Mallet-Paret, and J. A. Yorke, A bifurcation invariant: degenerate orbits treated as clusters of simple orbits, Lecture Notes in Math., vol. 1007, Springer-Verlag, Berlin, 1983, pp. 109-131. MR 730267 (85d:58058)
  • [11] K. T. Alligood, J. Mallet-Paret, and J. A. Yorke, An index for the global continuation of relatively isolated sets of periodic orbits, Lecture Notes in Math., vol. 1007, Springer-Verlag, Berlin, 1983, pp. 1-21. MR 730259 (85b:58108)
  • [12] H. Poincaré, Mémoire sur les courbes définies par une equation différentiell, J. Mathematiques 7 (1881), 375-422; Oeuvre, Gauthier-Villar, Paris, 1880-1890.
  • [13] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Appl. Math. Sci., vol. 42. Springer-Verlag, New York, 1983. MR 709768 (85f:58002)
  • [14] J. Dieudonné, Foundations of modern analysis, Academic Press, New York, 1960. MR 0120319 (22:11074)
  • [15] E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955. MR 0069338 (16:1022b)
  • [16] M. W. Hirsch and S. Smale, Differential equations, dynamical systems and linear algebra, Academic Press, New York, 1974. MR 0486784 (58:6484)
  • [17] L. M. Graves, The theory of functions of real variables, McGraw-Hill, New York, 1956. MR 0075256 (17:717f)
  • [18] A. A. Andronov et al., Qualitative theory of second order dynamical systems, Wiley, New York, 1973.
  • [19] I. Bendixson, Sur les courbes définies par des equations différentielles, Acta Math. 24 (1901), 1-88. MR 1554923
  • [20] L. M. Perko and S. S. Lung, Existence, uniqueness, and nonexistence of limit cycles for a class of quadratic systems in the plane, J. Differential Equations 53 (1984), 146-171. MR 748237 (85f:34057)
  • [21] B. Coll. A. Gasull, and J. L. Libre, Some theorems on the existence, uniqueness, and nonexistence of limit cycles for quadratic systems, J. Differential Equations 67 (1987), 372-399. MR 884276 (88c:34038)
  • [22] L. M. Perko, Bifurcation of limit cycles, Proc. Conference on Bifurcations and Periodic Orbits of Planar Vector Fields, Luminy, France, September 1989; Lecture Notes in Math., Springer-Verlag, (to appear). MR 1094385 (92c:58111)
  • [23] L. M. Perko, Bifurcation of limit cycles: geometric theory, Proc. Amer. Math. Soc. (submitted). MR 1086341 (92c:34046)

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: https://doi.org/10.1090/S0002-9947-1990-0998357-4
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society