Global Bifurcations of Limit Cycles
Author(s):
V. A. Gaiko
Belarussian State University of Informatics and Radioelectronics
Department of Mathematics
Belarus, 220090 Minsk, Koltsov Str.49–305
vlrgk@cit.org.by
Abstract:
Two-dimensional polynomial dynamical systems are considered. The
main problem of the qualitative theory of such systems is
Hilbert's Sixteenth Problem:
Problem. Find the maximum number and relative position
of limit cycles.
There are three bifurcations of limit cycles:
Andronov-Hopf bifurcation (from a singular point of the center
or focus type);
separatrix cycle bifurcation (from a homoclinic or heteroclinic orbit);
multiple limit cycle bifurcation.
All these bifurcations are local ones. We consider only a neighborhood of
either the point or the separatrix cycle, or the multiple limit
cycle studying only the corresponding sufficiently small
neighborhood in the parameter space. The analysis of each of
these bifurcations individually does not yield a complete solution
of the Problem even in the simplest (quadratic) case of nonlinear
polynomial systems. We connect all local bifurcations of limit
cycles, develop a global bifurcation theory of polynomial
dynamical systems and on the example of quadratic systems outline
a global approach to the solution of Hilbert's 16th Problem. We
discuss also possibilities of application of the obtained results
to higher-dimensional dynamical systems and for the global
qualitative investigation of mathematical models from
microelectronics and ecology.
First, we recall some previous results. Using Erugin's ideas on
the qualitative investigation on the whole (both on the whole
phase plane and on the whole parameter space) and his two-isocline
method, we construct canonical systems with field-rotation
parameters and give a geometric interpretation of all four cases
of center for the quadratic systems:
1) axial symmetry;
2) local symmetry (zero divergence) on the whole phase plane (Hamiltonian
case);
3) orthogonality of asymptotes of hyperbolas forming
the family of isoclines (Lotka-Volterra case);
4) orthogonality of asymptotes of saddles at infinity.
Basing on the center cases
and applying the field-rotation parameters, we carry out the
classification of separatrix cycles with the corresponding
division of the parameter space and control the multiple limit
cycle bifurcations.
Then, using the Wintner-Perko termination principle, we give a
sketch of proof of the following conjecture:
Conjecture 1. There exists no quadratic system having a
swallow-tail bifurcation surface of multiplicity-four limit
cycles in its parameter space. In other words, a quadratic system
cannot have neither a multiplicity-four limit cycle nor four limit
cycles around a singular point (focus) and the maximum
multiplicity or the maximum number of limit cycles surrounding a
focus is equal to three.
The proof is carried out by contradiction. We suppose that our
canonical system containing three field-rotation parameters has
four limit cycles around the origin; then we get into some
three-dimensional domain of the field-rotation parameters being
restricted by some conditions on the rest two parameters
corresponding to the definite case of singular points in the
phase plane. This three-parameter domain of four limit cycles is
bounded by three fold bifurcation surfaces forming a swallow-tail
bifurcation surface of multiplicity-four limit cycles. It can be
shown that the corresponding maximal one-parameter family of
multiplicity-four limit cycles cannot be cyclic and terminates
either at the origin or on some separatrix cycle surrounding the
origin. Since we know absolutely precisely at least the cyclicity
of the singular point (the result by N.N.Bautin) which is
equal to three, we have got a contradiction with the termination
principle stating that the multiplicity of limit cycles cannot be
higher than the multiplicity (cyclicity) of the singular point in
which they terminate. This contradiction concludes the proof.
Since we know the concrete properties of all three field-rotation
parameters in the canonical system and, besides, we are able to
control simultaneously bifurcations of limit cycles around
different singular points, we can formulate also
Conjecture 2. The maximum number of limit cycles in a
quadratic system is equal to four and the only possible their
distribution is (3:1).
