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Novel algebraic aspects of Liouvillian integrability for two-dimensional polynomial dynamical systems
Physics Letters A. 2018. Vol. 382. No. 20. P. 1353-1360.
The general structure of irreducible invariant algebraic curves for a polynomial dynamical system in C^2 is found. Necessary conditions for existence of exponential factors related to an invariant algebraic curve are derived. As a consequence, all the cases when the classical force-free Duffing and Duffing–van der Pol oscillators possess Liouvillian first integrals are obtained. New exact solutions for the force-free Duffing–van der Pol system are constructed.
Demina M.V., Analysis and Mathematical Physics 2021 Vol. 11 No. 1 P. 25
The problem of Liouvillian integrability for the classical force-free generalized Duffing
oscillators is solved completely. All the cases when the generalized Duffing oscillators
possess Liouvillian first integrals are classified. It is shown that the general solutions
in integrable cases are expressible via elliptic and hyperelliptic functions. The relationship
between the generalized Duffing systems and the Newell–Whitehead–Segel
equation is used to ...
Added: January 4, 2021
Demina M.V., Applied Mathematics Letters 2018 Vol. 84 P. 42-48
A novel algebraic method for finding invariant algebraic curves for a polynomial vector field in $\mathbb{C}^2$ is introduced. The structure of irreducible invariant algebraic curves for Li\'{e}nard dynamical systems $x_t=y$, $y_t=-g(x)y-f(x)$ with $\text{deg} f=\text{deg} g+1$ is obtained. It is shown that there exist Li\'{e}nard systems that possess more complicated invariant algebraic curves than it was ...
Added: September 29, 2018
Demina M.V., Communications in Contemporary Mathematics 2022 Vol. 24 No. 3 Article 2150007
An explicit expression for the cofactor related to an irreducible invariant
algebraic curve of a polynomial dynamical system in the plane is derived. A
sufficient condition for a polynomial dynamical system in the plane to have a
finite number of irreducible invariant algebraic curves is obtained. All
these results are applied to Lienard dynamical systems $x_t=y$,
$y_t=-f(x)y-g(x)$ with $\deg f<\deg ...
Added: June 15, 2021
Demina M.V., Journal of Physics A: Mathematical and Theoretical 2018 Vol. 51 No. 505202 P. 1-17
A novel method of finding and classifying irreducible invariant surfaces of non-autonomous polynomial dynamical systems in the plane is presented. The general structure of irreducible invariant surfaces and their cofactors is found. The complete set of irreducible invariant surfaces for the classical forced Duffing-van der Pol oscillator is obtained. It is proved that the forced ...
Added: November 1, 2018
Demina M.V., Sinelshchikov D.I., Symmetry 2019 Vol. 11 No. 11 P. 1-10
We consider a family of cubic Liénard oscillators with linear damping. Particular cases of this family of equations are abundant in various applications, including physics and biology. There are several approaches for studying integrability of the considered family of equations such as Lie point symmetries, algebraic integrability, linearizability conditions via various transformations and so on. ...
Added: November 12, 2019
Demina M.V., Kuznetsov N.S., Journal of Dynamical and Control Systems 2021 Vol. 27 No. 2 P. 403-415
The upper bound on the degrees of irreducible Darboux polynomials associated
to the ordinary differential equations $ x_{tt}+\varepsilon {x_t}^2+\eta
x_t+f(x)=0 $ with $ f(x)\in\mathbb{C}[x]\setminus\mathbb{C} $ and $
\varepsilon\neq0 $ is derived. The availability of this bound provides the
solution of the Poincar\'{e} problem. Results on uniqueness and existence of
Darboux polynomials are presented. The problem of Liouvillian integrability
for related dynamical ...
Added: September 23, 2020
Demina M.V., Studies in Applied Mathematics 2023 Vol. 150 No. 3 P. 755-817
We provide the necessary and sufficient conditions of Liouvillian integrability for Liénard differential systems describing nonlinear oscillators with a polynomial damping and a polynomial restoring force. We prove that Liénard differential systems are not Darboux integrable excluding subfamilies with certain restrictions on the degrees of the polynomials arising in the systems. We demonstrate that if ...
Added: May 12, 2023
Demina M.V., Electronic Journal of Qualitative Theory of Differential Equations 2021 Vol. 48 P. 1-22
We present a set of conditions enabling a polynomial system of ordinary differential equations in the plane to have invariant algebraic curves. These conditions are necessary and sufficient. Our main tools include factorizations over the field of Puiseux series near infinity of bivariate polynomials generating invariant algebraic curves. The set of conditions can be algorithmically ...
Added: September 21, 2021
Kazakov A., Леванова Т. А., Коротков А. Г. et al., Известия высших учебных заведений. Прикладная нелинейная динамика 2018 Т. 26 № 5 С. 101-112
The phenomenological model of an ensemble of three neurons which are coupled by
chemical (synaptic) and electrical couplings is studies. A neuron is modeled by the oscillator of van
der Pol. The aim of work is a study of the influence of coupling’s strength and frequency detuning
between elements at regime of sequential activity that is observed in ...
Added: October 26, 2018
Жеглов А. Б., Osipov D., Сибирский математический журнал 2019 Т. 60 № 4 С. 760-776
In the paper, Lax pairs for linear Hamiltonian systems of differential equations are constructed. In particular, Groebner bases are used for the computations. It is proved that the maps which appear in the construction of Lax pairs are Poisson. Various properties of first integrals of the system which are obtained from the Lax pairs are ...
Added: October 8, 2019
Demina M.V., Discontinuity, Nonlinearity, and Complexity 2020 Vol. 9 No. 4 P. 499-507
We solve completely the problem of Liouvillian integrability for cubic
Lienard differential equations with quadratic damping. Our main tool is
the method of Puiseux series. We find necessary and sufficient conditions
for equations under consideration to have Jacobi last multipliers of a special
form. It turns out that some particular sub–families being Liouvillian non–
integrable possess Jacobi last multipliers. The ...
Added: August 24, 2020
Gordin V. A., М. : Физматлит, 2013
Описаны аналитические и численные методы исследования уравнений и систем в частных производных: гиперболических, параболических, эллиптических и смешанного типа, линейных и нелинейных. Список этих методов и приемов велик, и они должны дополнять друг друга: интегральные преобразования, вариационное исчисление, специальные функции, асимптотические методы, сплайны, рациональные аппроксимации.
Книга адресована читателю, который использует и аналитические, и численные, компьютерные ...
Added: March 18, 2013
Demina M.V., Valls C., International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2020 Vol. 30 No. 4 P. 2050056-1-2050056-9
We give the complete classication of irreducible invariant algebraic curves in family (I) of the
Chinese classication. In addition, we provide a complete and correct proof of the non{existence of
algebraic limit cycles for these equations. ...
Added: October 22, 2019
Demina M.V., Valls C., Proceedings of the Royal Society of Edinburgh: Section A 2020 Vol. 150 No. 6 P. 3231-3251
We give the complete classification of irreducible invariant algebraic curves of quadratic Lienard differential equations.
We prove that these equations have irreducible invariant algebraic curves of unbounded degrees, in contrast with what is wrongly claimed in the literature. In addition, we classify all the quadratic Lienard differential equations that admit a Liouvillian first integral. ...
Added: October 22, 2019
Demina M.V., Sinelshchikov D., Journal of Geometry and Physics 2021 Vol. 165 P. 104215-1-104215-12
Nonlinear oscillators described by polynomial Liénard differential equations arise in a variety of mathematical and physical applications. For a family of generalized Duffing–van der Pol oscillators we classify Darboux integrable cases and explicitly construct the corresponding generalized Darboux first integrals. We demonstrate that Darboux integrability is in strong correlation with the linearizability via the generalized ...
Added: May 27, 2021
Жеглов А. Б., Osipov D., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2018 Т. 483 № 5 С. 482-484
Lax pairs for linear Hamiltonian systems of differential equations are found in the paper. The first integrals of the system obtained from these Lax pairs are investigated. ...
Added: October 5, 2018
Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189
The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...
Added: January 28, 2020
Borzykh D., ЛЕНАНД, 2021
Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...
Added: February 20, 2021
В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72-80
Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...
Added: May 3, 2017
Красноярск : ИВМ СО РАН, 2013
Труды Пятой Международной конференции «Системный анализ и информационные технологии» САИТ-2013 (19–25 сентября 2013 г., г.Красноярск, Россия): ...
Added: November 18, 2013
Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18
Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...
Added: November 16, 2020
Grines V., Gurevich E., Pochinka O., Russian Mathematical Surveys 2017 Vol. 71 No. 6 P. 1146-1148
In the paper a Palis problem on finding sufficient conditions on embedding of Morse-Smale diffeomorphisms in topological flow is discussed. ...
Added: May 17, 2017
Okounkov A., Aganagic M., Moscow Mathematical Journal 2017 Vol. 17 No. 4 P. 565-600
We associate an explicit equivalent descendent insertion to any relative insertion in quantum K-theory of Nakajima varieties.
This also serves as an explicit formula for off-shell Bethe eigenfunctions for general quantum loop algebras associated to quivers and gives the general integral solution to the corresponding quantum Knizhnik Zamolodchikov and dynamical q-difference equations. ...
Added: October 25, 2018
Danilov B.R., Moscow University Computational Mathematics and Cybernetics 2013 Vol. 37 No. 4 P. 180-188
The article investigates a model of delays in a network of functional elements (a gate network) in an arbitrary finite complete basis B, where basis elements delays are arbitrary positive real numbers that are specified for each input and each set of boolean variables supplied on the other inputs. Asymptotic bounds of the form τ ...
Added: December 2, 2019