Field-rotation parameters and monotonic families of multiple limit cycles
by
Valery A. Gaiko
Belarusian State University of Informatics and Radioelectronics

We consider one-parameter families of two-dimensional polynomial dynamical systems with field-rotation parameters and use Perko's theorem stating that if there exists a non-singular multiple limit cycle in a system, then it belongs to a one-parameter family of limit cycles of this system; furthermore: 1) if the multiplicity of the cycle is odd, then the family either expands or contracts monotonicaly under the variation of a field-rotation parameter; 2) if the multiplicity of the cycle is even, then it befurcates into a stable and an unstable limit cycle as the parameter varies in one sense and it disappears as the parameter varies in the opposite sense.

Applying this theorem for the quadratic systems and also using Bautin's result on the cyclicity of a singular point which is equal to three in this case and the Wintner-Perko termination principle stating that the multiplicity of limit cycles cannot be higher than the multiplicity (cyclicity) of the singular point in which they terminate, by means of a canonical system with three field-rotation parameters we prove by contradiction the non-existence of a quadratic system having a swallow-tail bifurcation surface of multiplicity-four limit cycles in its parameter space; in other words, we prove that a quadratic system cannot have neither a multiplicity-four limit cycle nor four limit cycles around a singular point (focus) and that both the maximum multiplicity and the maximum number of limit cycles surrounding a focus are equal to three.

Date received: May 22, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calh-65.