The Tangential Center-Focus Problem

Part of the Advanced Courses in Mathematics CRM Barcelona book series (ACMBIRK)


As is well known, the second part of Hilbert’s 16th problem is concerned with bounding the number of limit cycles in a polynomial system (1.2) of degree n in terms of n. This is a very hard problem, but Arnold has suggested a “Weak Hilbert’s 16th problem” which seems far more tractable: to find a bound on the number of limit cycles which can bifurcate from a first-order perturbation of a Hamiltonian system,
$$ \dot x = - \frac{{\partial H}} {{\partial y}} + \varepsilon P,\dot y = \frac{{\partial H}} {{\partial x}} + \varepsilon Q, $$
where the Hamiltonian, H, is a polynomial of degree n + 1 and the perturbation terms, P and Q are polynomials of degree m.


Riemann Surface Homology Group Polynomial System Perturbation Term Dehn Twist 
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© Birkhäuser Verlag 2007

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