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## Уточные циклы в типичных быстро-медленных системах на торе

Transactions of the Moscow Mathematical Society. 2010. № 71. С. 200-234.

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

Schurov I., Солодовников Н. А., Duck factory on the two-torus: multiple canard cycles without geometric constraints / Cornell University. Series math "arxiv.org". 2014. No. 1405.3251.

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 Poincare map is integer and the slow curve is connected, the number of canard limit cycles is ...

Added: May 14, 2014

Glutsyuk A., Bibilo Y., On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation / Cornell University. Series arXiv "math". 2021. No. 2011.07839.

We study family of dynamical systems on 2-torus modeling over-damped Josephson junction in superconductivity. It depends on three parameters (B,A;ω): B (abscissa), A(ordinate), ω (a fixed frequency).We study the rotation numberρ(B,A;ω) as a function of (B,A) withfixedω. Aphase-lock areais the level set Lr:={ρ=r}, if it has an on-empty interior. This holds for r∈Z (a result ...

Added: November 26, 2020

Buff X., Goncharuk N. B., Journal of Modern Dynamics 2015 Vol. 9 P. 169-190

We investigate the notion of complex rotation number which was introduced by V.I.Arnold in 1978. Let f: R/Z -> R/Z be a (real) analytic orientation preserving circle diffeomorphism and let omega in C/Z be a parameter with positive imaginary part. Construct a complex torus by glueing the two boundary components of the annulus { z ...

Added: October 10, 2013

Romaskevich O. L., Клепцын В. А., Schurov I., Наноструктуры. Математическая физика и моделирование 2013 Т. 8 № 1 С. 31-46

In order to model the processes taking place in systems with Josephson contacts, a differential equation on a torus with three parameters is used. One of the parameters of the system can be considered small and the methods of the fast-slow systems theory can be applied. The properties of the phase-lock areas - the subsets ...

Added: December 25, 2012

Klimenko A. V., Romaskevich O. L., Moscow Mathematical Journal 2014 Vol. 14 No. 2 P. 367-384

A three-parametrical family of ODEs on a torus arises from a model of Josephson effect in a resistive case when a Josephson junction is biased by a sinusoidal microwave current. We study asymptotics of Arnold tongues of this family on the parametric plane (the third parameter is fixed) and prove that the boundaries of the ...

Added: September 5, 2014

Schurov I., Клепцын В. А., Romaskevich O. L., Наноструктуры. Математическая физика и моделирование 2013 Т. 8 № 1 С. 31-46

In order to model the processes taking place in systems with Josephson contacts, a differential equation on a torus with three parameters is used. One of the parameters of the system can be considered small and the methods of the fast-slow systems theory can be applied. The properties of the phase-lock areas – the subsets ...

Added: December 17, 2014

Schurov I., Клепцын В. А., Romaskevich O. L., Наноструктуры. Математическая физика и моделирование 2014 Т. 8 № 1 С. 31-46

In order to model the processes taking place in systems with Josephson contacts, a differential equation on a torus with three parameters is used. One of the parameters of the system can be considered small and the methods of the fast-slow systems theory can be applied. The properties of the phase-lock areas – the subsets ...

Added: December 25, 2014

Glutsyuk A., Journal of Dynamical and Control Systems 2019 Vol. 25 No. 3 P. 323-349

In 1973, B. Josephson received a Nobel Prize for discovering a new fundamentaleffect concerning a Josephson junction,—a system of two superconductors separated by a very narrow dielectric: there could exist a supercurrent tunneling through this junction. We will discuss the model of the overdamped Josephson junction, which is given by a family of first-order nonlinear ...

Added: August 20, 2018

Filimonov D., Дифференциальные уравнения 2010 Т. 46 № 5 С. 647-657

В настоящей работе предложен метод исследования, позволяющий локализовать предель-ные циклы для векторных полей на плоскости, содержащих сверхмедленный фокус. Этим методом исследована большая область на фазовой плоскости для уравнения Ши Сонглина. Предельные циклы в окрестности особой точки (0, 0) локализованы в узких кольцах, где доказана их единственность. ...

Added: November 14, 2013

Volk D., Kleptsyn V., Gorodetski A. et al., Moscow Mathematical Journal 2014 Vol. 14 No. 2 P. 291-308

We consider a minimal action of a finitely generated semigroup by homeomorphisms of the circle, and show that the collection of translation numbers of individual elements completely determines the set of generators (up to a common continuous change of coordinates). One of the main tools used in the proof is the synchronization properties of random ...

Added: December 30, 2015

On determinants of modified Bessel functions and entire solutions of double confluent Heun equations

Buchstaber V., Glutsyuk A., Nonlinearity 2016 Vol. 29 No. 12 P. 3857-3870

We investigate the question on existence of entire solutions of well-known linear differential equations that are linearizations of nonlinear equations modeling the Josephson effect in superconductivity. We consider the modified Bessel functions Ij(x) of the first kind, which are Laurent series coefficients of the analytic function family . For every we study the family parametrized by , , of -matrix functions formed ...

Added: June 17, 2021

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

Бухштабер В. М., Glutsyuk A., Труды Математического института им. В.А. Стеклова РАН 2017 Т. 297 С. 62-104

Abstract—We study a family of double confluent Heun equations of the form LE = 0, where
L = L(λ,μ,n) is a family of second-order differential operators acting on germs of holomorphic
functions of one complex variable. They depend on complex parameters λ, μ, and n. The
restriction of the family to real parameters satisfying the inequality λ + μ^2>0 ...

Added: June 29, 2018

Kruglov V., Pochinka O., Таланова Г. Н., Proceedings of the International Geometry Center 2020 Vol. 13 No. 1 P. 49-60

Currently, a complete topological classification has been obtained with respect to the topological equivalence of Morse-Smale flows, [9,7], as well as their generalizations of Ω-stable flows on closed surfaces, [4]. Some results on topological conjugacy classification for such systems are also known. In particular, the coincidence of the classes of topological equivalence and conjugacy of ...

Added: June 28, 2020

Kazakov A., Lerman L. M., Kulagin N., Mathematical Modeling of Natural Phenomena 2013 Vol. 8 No. 5 P. 155-172

We demonstrate that a piecewise linear slow-fast Hamiltonian system with an equilibrium of the saddle-center type can have a sequence of small parameter values for which a one-round homoclinic orbit to this equilibrium exists. This contrasts with the well-known findings by Amick and McLeod and others that solutions of such type do not exist in ...

Added: March 29, 2015

Romanov A., Kondratieva L., Inertial Manifolds and Limit Cycles of Dynamical Systems in Rn / Cornell University. Series math "arxiv.org". 2019. No. 1911.03932.

We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in Rn permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincare–Bendixson theory. In the
case n = 3 we implement such a scenario for a model of a satellite rotation around a celestial body of small ...

Added: November 13, 2019

Chistyakova S. A., Dolov M. V., Differential Equations 2012 Vol. 48 No. 8 P. 1180-1182

For a certain class of two-dimensional autonomous systems of differential equations with an invariant curve that contains ovals, we indicate necessary and sufficient conditions for these ovals to be limit cycles of phase trajectories. ...

Added: March 15, 2013

Bizyaev I. A., Borisov A. V., Killin A. A. et al., Regular and Chaotic Dynamics 2016 Vol. 21 No. 6 P. 759-774

This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector (3, 6, 14), the other is defined by two generatrices and growth vector (2, 3, 5, 8). Using a Poincar´e map, the nonintegrability of the above systems in the general ...

Added: April 5, 2017

Bizyaev I. A., Borisov A. V., Killin A. A. et al., Regular and Chaotic Dynamics 2016 Vol. 21 No. 6 P. 759-774

This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector (3, 6, 14), the other is defined by two generatrices and growth vector (2, 3, 5, 8). Using a Poincar´e map, the nonintegrability of the above systems in the general ...

Added: April 4, 2017

Kondratieva L., Romanov A., Electronic Journal of Qualitative Theory of Differential Equations 2019 No. 96 P. 1-11

We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in Rn permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincaré–Bendixson theory. In the case n = 3 we implement such a scenario for a model of a satellite rotation around a celestial body of ...

Added: December 22, 2019

Ilyashenko Y., Известия РАН. Серия математическая 2016 Т. 80 № 1 С. 55-118

Настоящая работа является первой из двух частей, в которых излагает- ся дайджест доказательства теоремы конечности для предельных циклов полиномиального векторного поля на плоскости. В то же время дают- ся схемы доказательств еще двух теорем: аналогичного результата для аналитических векторных полей и описания асимптотики преобразования монодромии для сложных циклов таких полей. ...

Added: March 4, 2016

Golikova L., Зинина С. Х., Известия высших учебных заведений. Прикладная нелинейная динамика 2021 Т. 29 № 6 С. 851-862

It is known from the 1939 work of A. G. Mayer that rough transformations of the circle are limited to the
diffeomorphisms of Morse – Smale. A topological conjugacy class of orientation-preserving diffeomorphism is entirely determined by its rotation number and the number of its periodic orbits, while for orientation-changing diffeomorphism the topological invariant will be ...

Added: December 3, 2021

On determinants of modified Bessel functions and entire solutions of double confluent Heun equations

Glutsyuk A., Buchstaber V., Nonlinearity 2016 Vol. 29 No. 12 P. 3857-3870

We investigate the question on existence of entire solutions of well-known linear differential
equations that are linearizations of nonlinear equations modeling the Josephson effect in superconductivity.
We consider the modified Bessel functions $I_j(x)$ of the first kind, which are Laurent series coefficients of the analytic
function family $e^{\frac x2(z+\frac 1z)}$. For every $l\geq1$ we study the family parametrized ...

Added: February 16, 2017