Why limit cycles?
have problems describing my field of interests. To put it simply, I love
ordinary differential equations, especially the analytic theory of ODE’s. Unfortunately, today it sounds ambiguous and
not fashionable. Yet I believe that the central position of this area, on the
cross-roads of many paths, overweighs these considerations.
specifically, what I am now interested in, is a number of problems spiritually
related to the Hilbert 16th problem, one of the two last problems
from the Hilbert’s list of 1901, which still remain open (the other one
is the Riemann Hypothesis on zeta-function). Hilbert asked, what can be the
number and position of Poincare limit
cycles, isolated closed (periodic)
solutions of a polynomial differential equation on the plane of the form P(x,y)dx+Q(x,y)dy=0.
Such question, if posed for very similar object, closed (compact) ovals
of an algebraic curve H(x,y)=0, would be very natural (in fact,
it is the first part of the same 16th problem). The problem is that
limit cycles almost never are algebraic curves, hence counting them is a
completely different problem belonging to a totally different realm of
transcendental analysis. Asking about transcendental objects a question
appropriate for an algebraic context only, makes the problem transcendentally
difficult beyond all considerations. The most advanced achievement towards its
solution is the finiteness theorem (Yu. Ilyashenko,
J. Ecalle, 1991-1992) claiming that the number in
question is finite (in other words, that the question is well-formed!). Even
this statement requires a book-long proof.
a more basic level, the same phenomenon manifests itself in the following way.
Polynomial and more generally algebraic functions of one complex variable, and
their rational combinations, always admit upper bounds for the number of their
isolated zeros. The only operation that destroys algebraicity, is solving
differential equations (in particular, integration and exponentiation).
Solutions of polynomial differential equations may exhibit totally different
properties (compare real non-integer powers of a complex argument which inherit
many properties of usual monomials and the sine/cosine functions that have
infinitely many zeros). Analyzing
reasons for such (mis)behavior requires tools from so distant areas
as integral geometry, singularity theory and commutative algebra.
particular form of the “algebraic-like” question about
transcendental objects is known as the tangential or infinitesimal Hilbert
problem. Consider an infinite collection of real level curves H(x,y)=z, where z ranges over all
real values, and along each level curve integrate a polynomial differential
1-form P(x,y)dx+Q(x,y)dy. The result is a (multivalued) function of the argument z, called
Abelian integral. The problem is to find the number of “roots” of
this function, i.e., the number of algebraic ovals yielding zero values for
this integral. Despite its seemingly elementary character, the problem is also
open to a very large extent. For instance, only in March 2002 it was proved
that the number of such “roots” (ovals) can be at most 2 if all
polynomials H,P,Q are quadratic.
August 2008, together with Gal Binyamini and Dmitry Novikov, we finally
succeeded in solving this problem explicitly and proved that the number of isolated zeros of an Abelian integral does not exceed , where is the maximum of the degrees of and is an explicit polynomial of degree not exceeding, say, 60. This result on one hand settles a 50+ years-old problem, on the other hand, leads to a number of new intriguing questions.
and adjacent issues are explained in details in several lecture notes and
survey papers that I wrote on this subject. For more information you are
invited to browse the complete list
of (online) papers available on this site.
Yakovenko, S. Quantitative
theory of ordinary differential equations and tangential Hilbert 16th
problem, Preprint math.DS/0104140
(2001), 1--69, Lecture notes of the course delivered on the Workshop "Asymptotic
series, differential algebra and finiteness theorems" (Montreal,
2000), to appear in CRM Proceedings.
Yakovenko, S. On functions
and curves defined by ordinary differential equations. The Arnoldfest (Toronto, ON, 1997),
Fields Inst. Communications, 24 (1999), 497--525, Amer. Math. Soc.,
Providence, RI, 1999
report (1996) mentions and briefly surveys some of the previous work
Ilyashenko, Yu., Yakovenko, S. Concerning
the Hilbert sixteenth problem. Concerning the Hilbert
16th problem, 1--19, Amer. Math. Soc. Transl. Ser. 2, 165, Amer. Math. Soc.,
Providence, RI, 1995
hope to put here a list of problems that I for some reasons believe to be
important, interesting or promising.