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Linearizability conditions for Lotka-Volterra planar complex quartic systems having homogeneous nonlinearities. (English) Zbl 1217.34058

Summary: 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 \(x'=x(1-a_{30}x^3-a_{21}x^y-a_{12}xy^2-a_03y^3)\), \(y'=-y(1-b_{30}x^3-b_{21}x^2y-b_{12}xy^2-b_{03}y^3\). 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.

MSC:

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37G05 Normal forms for dynamical systems

Software:

primdec; SINGULAR
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