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Integrability conditions for complex homogeneous kukles systems. (English) Zbl 1417.34069

Summary: In this paper we study the existence of local analytic first integrals for complex polynomial differential systems of the form \(\dot{x} = x + P_n(x,y)\), \(\dot{y} = -y\), where \(P_n(x,y)\) is a homogeneous polynomial of degree \(n\), called the complex homogeneous Kukles systems of degree \(n\). We characterize all the homogeneous Kukles systems of degree \(n\) that belong to the Sibirsky ideal. Finally, we provide necessary and sufficient conditions when \(n = 2,\dots,7\) in order that the complex homogeneous Kukles system has a local analytic first integral computing the saddle constants and using Gröbner bases to find the decomposition of the algebraic variety into its irreducible components.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37C10 Dynamics induced by flows and semiflows
34C14 Symmetries, invariants of ordinary differential equations
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems

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SINGULAR; primdec
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