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  "body": "\n## Statement\n\nLet $P$ and $Q$ be real polynomials in two variables.\nThen the planar [[autonomous system|0673D85QKZ]]\n$$x' = P(x, y), \\qquad y' = Q(x, y)$$\nhas only finitely many [[limit cycles|0AAQ73WRFB]].\n\n## Remarks\n\nHenri Dulac published a proof in 1923, but a gap was found in it much later.\nComplete proofs were given independently by Yulij Ilyashenko and by Jean Écalle around 1991–1992.\nThe hard part is to exclude limit cycles that accumulate on a polycycle, a closed chain of equilibria joined by orbits.\nLimit cycles cannot accumulate on a periodic orbit of an analytic field, because the Poincaré return map near that orbit is analytic.\n\nThe theorem bounds the number of limit cycles of each single system.\nIt gives no bound in terms of the degree of $P$ and $Q$.\nWhether such a bound exists, even for degree $2$, is the second part of Hilbert's sixteenth problem, and it is open.\nFor example, the van der Pol system $x' = y - (x^3/3 - x)$, $y' = -x$ has exactly one limit cycle.\n\n## Explanations\n\n### L1 human\n\nA limit cycle is an isolated closed loop of motion in the plane.\nThe theorem says that a system given by polynomial formulas can only have finitely many of them.\nIt does not say how many, and nobody knows a general bound in terms of the degrees of the polynomials.\n\n### L2 human\n\nThe theorem is the finiteness part of the study of Hilbert's sixteenth problem on limit cycles.\nNear a periodic orbit, finiteness follows from the analyticity of the return map.\nThe difficulty lies at polycycles, where the return map has an asymptotic expansion that is not convergent.\nIlyashenko and Écalle overcame this with a theory of such expansions.\n",
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