The center problem for a 2:-3 resonant cubic Lotka–Volterra system

Abstract

In this paper we obtain conditions on the coefficients of a cubic Lotka–Volterra system of the form

equation(1) x=x(2-a20x2-a11xy-a02y2), ẏ=y(-3+b20x2+b11xy+b02y2),

which fulfillment yields the existence in a neighborhood of the origin of a first integral of the form ϕ(x,y)=x3y2+h.o.t.ϕ(x,y)=x3y2+h.o.t., in which case the origin is termed a 2:-32:-3 resonant center. This system was studied in [13], where, due to computational constrains, the consideration was limited to the cases where either one or both coefficients a11,b11a11,b11 in system (1) were equal to zero, or both coefficients were equal to 1. Here we are studying the case where the coefficient a11a11 is equal to 1 and b11b11 is arbitrary. The obtained results represent the study of the center problem for general system (1), since by a linear substitution any system of the form (1) can be transformed either to system (1) with a11=1a11=1 or to one of systems studied in [13]. Computation of the resonant saddle quantities (focus quantities) and the decomposition of the variety of the ideal generated by an initial string of them were used to obtain necessary conditions of integrability and the theory of Darboux first integrals and some other methods to show the sufficiency. Since the decompositions of the variety mentioned above was performed using modular computations the obtained 19 conditions of integrability represent the complete list of the integrability conditions only with very high probability and there remains an open problem to verify this statement.

The second author is partially supported by a MINECO/FEDER Grant No. MTM2011–22877 and by a AGAUR (Generalitat de Catalunya) Grant No. 2009SGR 381. The fourth author is supported by Slovenian Research Agency. This research was also supported by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme, FP7-PEOPLE-2012-IRSES-316338.