The centre and isochronicity problems for some cubic systems

We present an efficient method for computing focus and linearizability quantities of polynomial differential equation systems. We apply the method to computing these quantities for ten eight-parametric cubic systems and obtain the necessary and sufficient conditions of linearizability (isochronicity) of these systems. We also show that there is a kind of duality between the problem of constructing algebraic invariant curves, first integrals and linearizing transformations on one side, and the problem of solving some first-order linear partial differential equations on the other side.

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