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## Towards the General Theory of Global Planar Bifurcations

Ch. 13. P. 269-299.

This is an outline of a theory to be created, as it was seen in April 2015. An addendnum to the proofs at the end of the chapter describes the recent developments.

Keywords: векторные полябифуркацииbifurcationsvector fieldseparatrix connectionsсепаратрисные связки

Publication based on the results of:

### In book

Bourama T. Vol. 157. , NY : Springer, 2016

Yu. Ilyashenko, Kudryashov Y., I. Schurov, Inventiones Mathematicae 2018 Vol. 213 No. 2 P. 461-506

We construct an open set of structurally unstable three parameter families whose weak and so called moderate topological classification defined below has a numerical invariant that may take an arbitrary positive value. Here and below “families” are “families of vector fields in the two-sphere”. This result disproves an Arnold’s conjecture of 1985. Then we construct an ...

Added: February 6, 2018

Пригожин И., В кн. : Мир человека: неопределенность как вызов. : М. : ЛЕНАНД, 2019. С. 55-62.

Nobel prize winner I. Prigogine stands for peace, against the arms race, against the use of science for destruction of man and humanity. In his opinion, in the sphere of human capabilities it is essential to change the trajectory of civilization development. At the bifurcation points, unprecedented changes are possible. Instability is not a sign ...

Added: November 21, 2018

Ilyashenko Y., Solodovnikov N., Moscow Mathematical Journal 2018 Vol. 18 No. 1 P. 93-115

Global bifurcations in the generic one-parameter families that unfold a vector field with a separatrix loop on the two-sphere are described. The sequence of bifurcation that occurs is in a sense in ono-to-one correspondence with finite sets on a circle having some additional structure on them. Families under study appear to be structurally stable. The ...

Added: December 15, 2017

Burov A. A., Якушев И. А., Прикладная математика и механика 2014 Т. 78 № 5 С. 645-655

Рассматривается скольжение тяжелой бусинки, нанизанной на тонкий круговой обруч, вращающийся с постоянной угловой скоростью вокруг вертикальной оси, расположенной в его плоскости и, в общем случае, не проходящей через его вертикальный диаметр. Предполагается, что между бусинкой и обручем действует сила сухого трения. Находятся множества неизолированных положений относительного равновесия бусинки на обруче, исследуется их зависимость от параметров ...

Added: November 27, 2014

Ilyashenko Y., Kudryashov Y., Schurov I., / Cornell University. Series math "arxiv.org". 2015. No. 1506.06797.

This is the first part of a two parts paper dedicated to global bifurcations in the plane. In this part we construct an open set of three parameter families whose topological classification has a numerical invariant that may take an arbitrary positive value. In the second part we construct an open set of six parameter ...

Added: June 24, 2015

Schurov I., Solodovnikov N., Journal of Dynamical and Control Systems 2017 Vol. 23 No. 3 P. 481-498

Slow-fast systems on the two-torus are studied. As it was shown before, canard cycles are generic in such systems, which is in drastic contrast with the planar case. It is known that if the rotation number of the Poincaré map is an integer and the slow curve is connected, the number of canard limit cycles ...

Added: July 17, 2016

Korotkov A., Levanova T., Zaks M. et al., Communications in Nonlinear Science and Numerical Simulation 2022 No. 104 Article 106045

A minimalistic model of the half-center oscillator is proposed. Within it, we consider dynamics of two excitable neurons interacting by means of the excitatory coupling. In the parameter space of the model, we identify the regions of dynamics, characteristic for central pattern generators: respectively, in-phase, anti-phase synchronous oscillations and quiescence, and study various bifurcation transitions ...

Added: October 25, 2021

Гончарук Н. Б., Ilyashenko Y., Труды Математического института им. В.А. Стеклова РАН 2020 Т. 310 С. 86-106

Обсуждаются различные определения эквивалентности для бифуркаций векторных полей на сфере, и приводится большое количество примеров (как известных, так и новых), которые иллюстрируют достоинства и недостатки разных определений. Кроме классических определений сильной и слабой эквивалентности, рассматриваются новые понятия Sing-эквивалентности и умеренной эквивалентности. Эти определения представляются более подходящими и соответствующими интуитивному понятию эквивалентных бифуркаций. Они были введены и использованы для описания структурной неустойчивости ...

Added: May 27, 2021

Ivan Shilin, / Cornell University. Series math "arxiv.org". 2019. No. arXiv:1903.01933.

An orientation-preserving non-contractible separatrix loop of a hyperbolic saddle of a vector field on a two-dimensional surface may be accumulated by a separatrix of the same saddle. We study the unfolding of such loops in generic one-parameter families of vector fields as a semi-local bifurcation. As a byproduct, we construct a countable family of pairwise ...

Added: November 12, 2020

Ilyashenko Y., Chaos 2021 Vol. 31 Article 013103

We study the geometry of the bifurcation diagrams of the families of vector fields in the plane. Countable number of pairwise non-equivalent germs of bifurcation diagrams in the two-parameter families is constructed. Previously, this effect was discovered for three parameters only. Our example is related to so-called saddle node (SN)–SN families: unfoldings of vector fields with one ...

Added: May 27, 2021

Springer, 2015

In the field of Dynamical Systems, nonlinear iterative processes play an important role. Nonlinear mappings can be found as immediate models for many systems from different scientific areas, such as engineering, economics, biology, or can also be obtained via numerical methods permitting to solve non-linear differential equations. In both cases, the understanding of specific dynamical ...

Added: September 4, 2017

Tyukhtina A. A., Kalinin A., Mathematical Methods in the Applied Sciences 2018 No. № № 18. V. 41 P. 9283-9292

Added: September 22, 2023

Pochinka O., Dolgonosova A., Круглов Е. В., Нелинейная динамика 2017 Т. 13 № 4 С. 573-578

In this paper, one of the possible scenarios for the creation of heteroclinic separators in the solar corona is described and realized. This reconnection scenario connects the magnetic field with two zero points of different signs, the fan surfaces of which do not intersect, with a magnetic field with two zero points which are connected ...

Added: October 17, 2017

http://www.shilnikov.unn.ru/en/news.html?id=20, 2020

International Conference "ShilnikovWorkshop-2020" dedicated to the memory of the outstanding Russian mathematician Leonid Pavlovich Shilnikov (1934-2011) will be held on 17-18 December, 2020 at the Lobachevsky State University of Nizhny Novgorod. The topics of the Conference include but not restricted by the following themes of the theory of dynamical systems: bifurcations, strange attractors, conservative and ...

Added: November 1, 2021

E. Yakovlev, Lerman L., Journal of Geometry and Physics 2019 Vol. 135 P. 70-79

A mathematically correct description is presented on the interrelations between the dynamics of divergence free vector fields on an oriented 3-dimensional manifold M and the dynamics of Hamiltonian systems. It is shown that for a given divergence free vector field Xwith a global cross-section there exist some 4-dimensional symplectic manifold M̃⊃M and a smooth Hamilton function H:M̃→R such that for some c∈R one gets M={H=c}and ...

Added: October 22, 2018

Gonchenko S. V., Gonchenko A. S., Kazakov A. et al., International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2018 Vol. 28 No. 11 P. 1830036-1-1830036-29

The paper is devoted to topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finite-dimensional smooth systems can exist in three different forms. This is dissipative chaos, the mathematical image of which is a strange attractor; conservative chaos, for which the ...

Added: October 26, 2018

Trifonov K., / Cornell University. Series arXiv "math". 2020. No. 3454820.

An analytic reversible Hamiltonian system with two degrees of freedom is studied in a neighborhood of its symmetric heteroclinic connection made up of a symmetric saddle-center, a symmetric orientable saddle periodic orbit lying in the same level of a Hamiltonian and two non-symmetric heteroclinic orbits permuted by the involution. This is a co- dimension one structure and therefore ...

Added: December 26, 2020

Kudryashova E., Leonov G. A., Kuznetsov N. V., IFAC-PapersOnLine 2015

In this paper an approach to modeling of the Tunisian social system in 2011–2014 is considered and the revolution, bifurcation, and controlled stabilization are discussed. Using statistical analysis of socio-economic indicators of Tunisia there are selected two bifurcation parameters, which have influenced on stability of socio-economic system of Tunisia. Based on this analysis the recommendations ...

Added: March 28, 2015

191574970, Sbornik Mathematics 1994 Vol. 185 No. 11 P. 3-22

Added: September 23, 2016

Gusein-Zade S., Алгебра и анализ 2021 Т. 33 № 3 С. 73-84

Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related with the Euler characteristic through the classical Poincar\'e--Hopf theorem. Generalized Euler characteristics (additive topological invariants of spaces with some additional structures) are sometimes related with corresponding analogues of indices of singular points. Earlier, there was defined ...

Added: May 2, 2021

Rudakov A. N., Шафаревич И., Успехи математических наук 1978 Т. 33 № 6(204) С. 231-232

Added: October 16, 2012

Arseyev P., Maslova N. S., Mantsevich V. N., Solid State Communications 2012 Vol. 152 P. 1545-1550

We analyzed theoretically localized charge relaxation in a double quantum dot (QD) system coupled with continuous spectrum states in the presence of localized electrons Coulomb interaction in a single QD. We have found that for a wide range of system parameters charge relaxation occurs through two stable regimes with significantly different relaxation rates. A peculiar ...

Added: October 28, 2014

Glutsyuk A., Rybnikov L. G., Nonlinearity 2017 Vol. 30 No. 1 P. 61-72

We consider two-parametric families of non-autonomous ordinary differential equations on the two-torus with coordinates (x, t) of the type x'=v(x)+A+Bf(t). We study its rotation number as a function of the parameters (A, B). The phase-lock areas are those level sets of the rotation number function that have non-empty interiors. Buchstaber, Karpov and Tertychnyi studied the ...

Added: February 15, 2017