Linearizability conditions for Lotka–Volterra planar complex quartic systems having homogeneous nonlinearities

Abstract

In this paper we investigate the linearizability problem for the two-dimensional Lotka–Volterra complex quartic systems which are linear systems perturbed by fourth degree homogeneous polynomials, i.e., we consider systems of the form x ̇ = x ( 1 − a 30 x 3 − a 21 x 2 y − a 12 x y 2 − a 03 y 3 ) , y ̇ = − y ( 1 − b 30 x 3 − b 21 x 2 y − b 12 x y 2 − b 03 y 3 ) . The necessary and sufficient conditions for the linearizability of this system are found. From them the conditions for isochronicity of the corresponding real system can be derived.