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Isochronicity of plane polynomial Hamiltonian systems. (English) Zbl 0949.34077
Polynomial Hamiltonian systems in the plane are studied which can have isochronous center singular points. More exactly, the author deals with a complex extension of the related system and closed vanishing cycles \(\gamma (h)\) on the complex Riemannian surface \(H^{-1}(h)=\{H=h\}\) being a complexification of the corresponding closed periodic orbit. Therefore, he does not distinct a saddle singular point and a center singular point, only their Morse property is essential. The first result of the paper is negative (but being expected): for some class of polynomials the system cannot have isochronous singular points. The second result consists in dintinguishing a class of polynomials for which \(\gamma (h)\) represents a zero homological cycle on the algebraic curve \(H^{-1}(h)\). Some connections between the existence of isochronous centers and the well known Jacobian conjecture is discussed.

MSC:
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
34M45 Ordinary differential equations on complex manifolds
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