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<big>'''Dynamical Systems'''</big>

Welcome to the official site of the seminar on Dynamical Systems supervised by Yu.S.Ilyashenko.

The seminar on Dynamical Systems supervised now by A.Gorodetski and Yu. Ilyashenko continues the activities started 30 years ago in the seminar on Differential Equations organized by N. Nekhoroshev and Yu. Ilyashenko.

The modern theory of dynamical systems is one of the main tools of natural studies. As a mathematical discipline it spreads from physics and probability to multidimentional complex analysis and algebraic geometry.

Some key problems are: How chaos occurs in deterministic systems? Where is the boundary between differential equations and probability theory in the description of the limit behavior of dynamical systems? Is our Solar System stable? What the attractor of a typical dynamical system looks like? What are the characteristic features of the foliations of a complex plane by analytic curves? What may be said about the number and location of limit cycles of a planar polynomial vector field? These questions (two of them going back to Poincare and Hilbert) are or were in the realm of the investigations of the seminar.

One of the traditions of the seminar is a part of a general tradition of Moscow Mathematical School. It is involving young students in the creative work on the very early stage of their education. Several new results are obtained by the third and second year undergraduate students participating the seminar.
== Current activities ==
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* [[Seminar talks 2020-2021]]
* [[Dynamical Systems 2019]]



== Papers ==
* [http://www.dyn-sys.org/w/images/3/35/CHA20-AR-GBC2021-01328.pdf "Germs of bifurcation diagrams and SN-SN families"]  by Yu. Ilyashenko, Chaos 31, 1, 2021, [https://doi.org/10.1063/5.0030742  https://doi.org/10.1063/5.0030742]
* Papers on global bifurcations in one-parameter families:
[http://www.dyn-sys.org/w/images/8/8c/Starichkova_tame_1_par_families.pdf "Global bifurcations in generic one-parameter families on S^2" by V. Starichkova.]

[http://www.dyn-sys.org/w/images/a/a6/IS_saddle_loop.pdf "Global bifurcations in generic one-parameter families with a separatrix loop on S^2" by Yu. Ilyashenko and N. Solodovnikov.]

[http://www.dyn-sys.org/w/images/5/5e/GIS_parabolic_cycle.pdf "Global bifurcations in generic one-parameter families with a parabolic cycle on S^2" by N. Goncharuk, Yu. Ilyashenko, and N. Solodovnikov.]    [https://arxiv.org/pdf/1707.09779.pdf (arXiv:1707.09779)]

* Papers on global bifurcations by V. Sh. Roitenberg (with his permission, in Russian):

[http://www.dyn-sys.org/w/images/c/c9/%D0%A0%D0%BE%D0%B9%D1%82%D0%B5%D0%BD%D0%B1%D0%B5%D1%80%D0%B3._%D0%9E_%D0%B1%D0%B8%D1%84%D1%83%D1%80%D0%BA%D0%B0%D1%86%D0%B8%D1%8F%D1%85_%D1%81%D0%B5%D0%BF%D0%B0%D1%80%D0%B0%D1%82%D1%80%D0%B8%D1%81_%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D1%8C%D0%BD%D1%8B%D1%85_%D0%BA_%D0%B4%D0%B2%D0%BE%D0%B9%D0%BD%D0%BE%D0%BC%D1%83_%D1%86%D0%B8%D0%BA%D0%BB%D1%83.pdf "О бифуркациях сепаратрис, предельных к двойному циклу", 1995]

[http://www.dyn-sys.org/w/images/c/cf/Roitenberg_thesis_2000.pdf "Нелокальные двухпараметрические бифуркации векторных полей на поверхностях", 2000]

[http://www.dyn-sys.org/w/images/4/42/%D0%A0%D0%BE%D0%B9%D1%82%D0%B5%D0%BD%D0%B1%D0%B5%D1%80%D0%B3._%D0%9E_%D0%B1%D0%B8%D1%84._%D0%B2.%D0%BF.%2C_%D0%B8%D0%BC%D0%B5%D1%8E%D1%89%D0%B8%D1%85_%D1%8F%D1%87%D0%B5%D0%B9%D0%BA%D1%83%2C_%D0%BE%D0%B3%D1%80%D0%B0%D0%BD%D0%B8%D1%87%D0%B5%D0%BD%D0%BD%D1%83%D1%8E_%D0%B4%D0%B2._%D1%86%D0%B8%D0%BA%D0%BB%D0%B0%D0%BC%D0%B8.pdf "О бифуркациях векторных полей, имеющих ячейку, ограниченную двойными циклами", 2008]

[https://www.gramota.net/articles/issn_1993-5552_2008_7_58.pdf "О бифуркациях векторных полей, имеющих ячейки, состоящие из траекторий, предельных к двойным циклам", 2008]

[https://www.gramota.net/articles/issn_1993-5552_2012_7_34.pdf "О бифуркациях векторных полей, имеющих сепаратрису, предельную к контуру из сепаратрис двух разноплановых седел", 2012]

[http://www.dyn-sys.org/w/images/5/5c/%D0%A0%D0%BE%D0%B9%D1%82%D0%B5%D0%BD%D0%B1%D0%B5%D1%80%D0%B3_%D1%84%D1%80%D0%B0%D0%B3%D0%BC%D0%B5%D0%BD%D1%82_%D0%BE_%D1%81%D0%B5%D0%BF%D0%B0%D1%80%D0%B0%D1%82%D1%80%D0%B8%D1%81%D0%B5_%D1%81%D0%B5%D0%B4%D0%BB%D0%BE%D1%83%D0%B7%D0%BB%D0%B0_%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D0%B9_%D0%BA_%D0%BF%D0%B5%D1%82%D0%BB%D0%B5_%D1%81%D0%B5%D0%B4%D0%BB%D0%B0.pdf "О некоторых нелокальных двухпараметрических бифуркациях векторных полей на поверхностях, имеющих особую точку типа седлоузла седел" (фрагмент)]

* Статья Диего М. Бенардета:

[https://www.tandfonline.com/doi/pdf/10.1080/00029890.2022.2005389 "Complex Methods for Bounds on the Number of Periodic Solutions with an Application to a Neural Model" by Diego M. Benardete (the paper expounds Yu. S. Ilyashenko's method of obtaining an upper bound on the number of limit cycles and applies this methos to a neural model)]

* Статья Ю. С. Ильяшенко про аттракторы галёркинских приближений уравнения Навье-Стокса:
[http://www.dyn-sys.org/w/images/a/af/Ilyashenko_-_SelMathSov.pdf "Weakly Contracting Systems and Attractors of Galerkin Approximations of Navier-Stokes Equations on the Two-Dimensional Torus", 1992]

== Links ==
* [http://www.dyn-sys.org/w/images/5/5f/GlobifF.pdf Slides] of Yu. Ilyashenko's talk on global bifurcations {{ref-ru}}
* [http://www.dyn-sys.org/public/ODE-notes/ODE.pdf ODE textbook: Bufetov, Goncharuk, Ilyashenko] (September 2, 2019). Current status is described [http://www.dyn-sys.org/public/ODE-notes/Status.doc here].
* [[Plan:Fall 2013]] {{ref-ru}}
* [[I-Ya translation]] {{ref-ru}}
* [[Российская_газета_о_НМУ|Russian gazette about IUM]] {{ref-ru}}
* [[Grant numbers]]