Linearizability and local bifurcation of critical periods in a cubic Kolmogorov system

摘要

Since Chicone and Jacobs investigated local bifurcation of critical periods for quadratic systems and Newtonian systems in 1989, great attention has been paid to some particular forms of cubic systems having special practical significance but less difficulties in computation. This paper is devoted to the linearizability and local bifurcation of critical periods for a cubic Kolmogorov system. We use the Darboux method to give explicit linearizing transformations for isochronous centers. Investigating the finite generation for the ideal of all period constants, which are of the polynomial form in six parameters, we prove that at most two critical periods can be bifurcated from the interior equilibrium if it is an isochronous center. Moreover, we prove that the maximum number of critical periods is reachable.