Small Divisors: Number Theory in Dynamical Systems

Jean-Christophe Yoccoz ()
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Jean-Christophe Yoccoz: Collège de France

A chapter in An Invitation to Mathematics, 2011, pp 43-54 from Springer

Abstract: Abstract We discuss dynamical systems with two or more moving particles, such as two planets orbiting around the sun. If the ratio of their rotation periods, say α, is rational, then the planets are in resonance, and the mutual interaction will make the dynamics unstable. If the period ratio α is irrational, it can be approximated arbitrarily well by rational numbers, and the stability depends on how good this approximation is in terms of the sizes of numerators and denominators. We discuss this in a mathematical model case that can be analyzed completely, the setting of iteration of quadratic polynomials z↦e 2πiα z+z 2, and show how this leads to questions of Diophantine approximation within number theory. Finally, we briefly mention the situation of more than two planets.

Keywords: Recurrence Relation; Planetary System; Irrational Number; Diophantine Approximation; Unperturbed System (search for similar items in EconPapers)
Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-19533-4_4

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DOI: 10.1007/978-3-642-19533-4_4

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