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Hilbert’s 16th Problem and Its Weak Form

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

Abstract

Consider the planar differential systems
$$ \dot x = P_n (x,y),\dot y = Q_n (x,y), $$
(1.1)
where Pn and Qn are real polynomials in x, y and the maximum degree of P and Q is n. The second half of the famous Hilbert’s 16th problem, proposed in 1900, can be stated as follows (see [70]):

For a given integer n, what is the maximum number of limit cycles of system (1.1) for all possible Pn and Qn ? And how about the possible relative positions of the limit cycles ?

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Copyright information

© Birkhäuser Verlag 2007

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