Integrability conditions for Lotka-Volterra planar complex quartic systems having homogeneous nonlinearities
The (complex) center-focus problem is investigated for the system \[ \begin{aligned} \dot x &= x(1- a_{30} x^3- a_{21} x^2y- a_{12} xy^2- a_{03} y^3),\\ \dot y &=-y(1- b_{30} x^3- b_{21} x^2y- b_{12} xy^2- b_{03} y^3).\end{aligned} \] Necessary conditions for a center, that is, conditions for the existence of a first integral of the form \(\Psi=xy+h\cdot o\cdot t\) are obtained. For all of them but one it is proved that they are also the sufficient conditions for the existence of such an integral. In the unsolved case, the system can be written as \[ \begin{aligned} \dot x &= x\Biggl(1- {3\over 16} x^3+ {1\over 2} x^2 y- xy^2+ {8\over 9} y^3\Biggr),\\ \dot y &-y\Biggl(1+ {9\over 16} x^3- x^2y+{5\over 3} xy^2+ {16\over 9} y^3\Biggr)\end{aligned} \] and by the substitution \(x\to ax\), \(y\to by\), where \(a= 2({2\over 3})^{1/3}\), \(b= ({3\over 2})^{2/3}\), the latter system is transformed to \[ \begin{aligned} \dot x &= x(1-(x- y)(x^2- xy+ 2y^2)),\\ \dot y &= y(-1-8x^3+ (5x+ 4y)(x^2- y^2)).\end{aligned} \]
scientific article; zbMATH DE number 6153123