Dana Schlomiuk, Nicolae Vulpe
Abstract:
The Lotka-Volterra planar quadratic differential systems have
numerous applications but the global study of this class proved to
be a challenge difficult to handle. Indeed, the four attempts to
classify them (Reyn (1987), Wöz-Buserkros (1993),
Georgescu (2007) and Cao and Jiang (2008)) produced results which
are not in agreement. The lack of adequate global classification
tools for the large number of phase portraits encountered,
explains this situation. All Lotka-Volterra systems possess
invariant straight lines, each with its own multiplicity. In this
article we use as a global classification tool for
Lotka-Volterra systems the concept of configuration of invariant
lines (including the line at infinity). The class splits according
to the types of configurations in smaller subclasses which makes
it easier to have a good control over the phase portraits in each
subclass. At the same time the classification becomes more
transparent and easier to grasp. We obtain a total of 112
topologically distinct phase portraits: 60 of them with exactly
three invariant lines, all simple; 27 portraits with invariant
lines with total multiplicity at least four; 5 with the line at
infinity filled up with singularities; 20 phase portraits of
degenerate systems. We also make a thorough analysis of the
results in the paper of Cao and Jiang [13]. In contrast
to the results on the classification in [13], done in
terms of inequalities on the coefficients of normal forms, we
construct invariant criteria for distinguishing these portraits
in the whole parameter space
of coefficients.
Submitted January 19, 2012. Published April 25, 2012.
Math Subject Classifications: 58K30, 34A26,34C05, 34C40.
Key Words: Quadratic vector fields; Lotka-Volterra differential systems;
phase portraits; affine invariant polynomials;
topological invariants
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Dana Schlomiuk Département de Mathématiques et de Statistiques Université de Montréal, Canada email: dasch@DMS.UMontreal.ca | |
Nicolae Vulpe Institute of Mathematics and Computer Science Academy of Science of Moldova, Moldova email: nvulpe@gmail.com |
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