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Linearizability conditions for Lotka-Volterra planar complex cubic systems. (English) Zbl 1188.34115

The authors give necessary and sufficient conditions for the linearizability of the two-dimensional complex cubic systems of the form ${x}^{\text{'}}=x\left(1+{P}_{1}\left(x,y\right)+{P}_{2}\left(x,y\right)\right),$ ${y}^{\text{'}}=-y\left(1+{Q}_{1}\left(x,y\right)+{Q}_{2}\left(x,y\right)\right),$ where ${P}_{k}$ and ${Q}_{k}$, $k=1,2,$ are homogeneous polynomials.

First several necessary conditions are obtained. This is done trough the computation of some quantities that give obstructions to the linearizability, called in the paper linearizability quantities.

The most difficult part is to prove that these conditions are in fact sufficient. This is achieved trough an extension of the usual Darboux theory of integrability. In this extension the linearizing changes of variables are obtained by using the invariant algebraic curves of the systems together with their first integrals and their inverse integrating factors. Finally, the results are applied to characterize the isochronous centers when ${P}_{k}$ and ${Q}_{k}$, $k=1,2$ are real polynomials.

##### MSC:
 34M25 Formal solutions, transform techniques (ODE in the complex domain) 37F75 Holomorphic foliations and vector fields 32S65 Singularities of holomorphic vector fields and foliations
##### Keywords:
Cubic system; linearization; isochronicity; Darboux theory
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