Персона: Кудряшов, Николай Алексеевич
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Институт лазерных и плазменных технологий
Стратегическая цель Института ЛаПлаз – стать ведущей научной школой и ядром развития инноваций по лазерным, плазменным, радиационным и ускорительным технологиям, с уникальными образовательными программами, востребованными на российском и мировом рынке образовательных услуг.
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Кудряшов
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Николай Алексеевич
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- ПубликацияОткрытый доступExact solutions of the equation for surface waves in a convecting fluid(2019) Kudryashov, N. A.; Кудряшов, Николай Алексеевич© 2018 Elsevier Inc. A method for finding exact solutions and the first integrals is presented. The basic idea of the method is to use the value of the Fuchs index that appears in the Painlevé test to construct the auxiliary equation for finding the first integrals and exact solutions of nonlinear differential equations. It allows us to obtain the first integrals and new exact solutions of some nonlinear ordinary differential equations. The main feature of the method is that we do not assign a solution function at the beginning, we find this function during calculations. This approach is conceptually equivalent to the third step of the Painlevé test and sometimes allows us to change this step. Our approach generalizes a number of other methods for finding exact solutions of nonlinear differential equations. We demonstrate a method for finding the traveling wave solutions and the first integrals of the well-known nonlinear evolution equation for description of surface waves in a convecting liquid. The general solution of this equation at some conditions on parameters and new traveling wave solutions of the fourth-order equation are found.
- ПубликацияОткрытый доступRemarks on the Fuchs indices and the first integrals for nonlinear ordinary differential equations(2019) Kudryashov, N. A.; Кудряшов, Николай Алексеевич© 2019 Published under licence by IOP Publishing Ltd.The Painlevé analysis of nonlinear ordinary differential equations is used to construct the first integrals. The connection between the Fuchs indices and the first integrals of nonlinear differential equations is discussed. Some simple prepositions are presented. Some first integrals for nonlinear ordinary differential are found taking into account the values of the Fuchs indexes.
- ПубликацияОткрытый доступMultistability in dynamics of an encapsulated bubble contrast agent: Coexistence of three attractors(2019) Garashchuk, I. R.; Sinelshchikov, D. I.; Kudryashov, N. A.; Кудряшов, Николай Алексеевич© 2019 Published under licence by IOP Publishing Ltd.In this work we discuss complex dynamics arising in a model describing behavior of an encapsulated bubble contrast agent oscillating close to an elastic wall. We demonstrate presence of three coexisting attractors in the system. We propose an efficient numerical procedure based on the continuation method that can be used to locate the area of coexistence of these attractors in the parameters space. We provide area of coexistence of three attractors obtained by means of the proposed procedure.
- ПубликацияОткрытый доступOn some properties of the coupled Fitzhugh-Nagumo equations(2019) Lavrova, S. F.; Kudryashov, N. A.; Sinelshchikov, D. I.; Лаврова, София Федоровна; Кудряшов, Николай Алексеевич© 2019 Published under licence by IOP Publishing Ltd.We consider the FitzHugh-Nagumo model describing two neurons electrically coupled via ion flow through gap juctions between them. This model is a simple example of a neural network, which has a vast amount of periodic behaviors. It is shown that system of equations describing this model does not pass the Painlevé test. Analysis of stability of system's trivial stationary point is carried out. It is shown that this equilibrium point is not always stable. For some parameter regions where solution oscillates bifurcation diagrams are plotted and Lyapunov exponents are calculated. It is shown that analyzed non-stationary solutions are quasiperiodic.
- ПубликацияОткрытый доступHyperchaos and multistability in the model of two interacting microbubble contrast agents(2019) Kazakov, A. O.; Garashchuk, I. R.; Sinelshchikov, D. I.; Kudryashov, N. A.; Кудряшов, Николай Алексеевич© 2019 Author(s).We study nonlinear dynamics of two coupled contrast agents that are micrometer size gas bubbles encapsulated into a viscoelastic shell. Such bubbles are used for enhancing ultrasound visualization of blood flow and have other promising applications like targeted drug delivery and noninvasive therapy. Here, we consider a model of two such bubbles interacting via the Bjerknes force and exposed to an external ultrasound field. We demonstrate that in this five-dimensional nonlinear dynamical system, various types of complex dynamics can occur, namely, we observe periodic, quasiperiodic, chaotic, and hypechaotic oscillations of bubbles. We study the bifurcation scenarios leading to the onset of both chaotic and hyperchaotic oscillations. We show that chaotic attractors in the considered system can appear via either the Feigenbaum cascade of period-doubling bifurcations or the Afraimovich-Shilnikov scenario of torus destruction. For the onset of hyperchaotic dynamics, we propose a new bifurcation scenario, which is based on the appearance of a homoclinic chaotic attractor containing a saddle-focus periodic orbit with its two-dimensional unstable manifold. Finally, we demonstrate that the dynamics of two bubbles can be essentially multistable, i.e., various combinations of the coexistence of the above mentioned attractors are possible in this model. These cases include the coexistence of a hyperchaotic regime with an attractor of any other remaining type. Thus, the model of two coupled gas bubbles provides a new example of physically relevant system with multistable hyperchaos.
- ПубликацияОткрытый доступLax representation and quadratic first integrals for a family of non-autonomous second-order differential equations(2019) Sinelshchikov, D. I.; Gaiur, I. Y.; Kudryashov, N. A.; Кудряшов, Николай Алексеевич© 2019 Elsevier Inc.We consider a family of non-autonomous second-order differential equations, which generalizes the Liénard equation. We explicitly find the necessary and sufficient conditions for members of this family of equations to admit quadratic, with the respect to the first derivative, first integrals. We show that these conditions are equivalent to the conditions for equations in the family under consideration to possess Lax representations. This provides a connection between the existence of a quadratic first integral and a Lax representation for the studied dissipative differential equations, which may be considered as an analogue to the theorem that connects Lax integrability and Arnold–Liouville integrability of Hamiltonian systems. We illustrate our results by several examples of dissipative equations, including generalizations of the Van der Pol and Duffing equations, each of which have both a quadratic first integral and a Lax representation.
- ПубликацияОткрытый доступStatistical features of plastic flow localization in materials(2019) Kudryashov, N. A.; Muratov, R. V.; Ryabov, P. N.; Кудряшов, Николай Алексеевич; Муратов, Родион Владимирович; Рябов, Павел Николаевич© 2019 Published under licence by IOP Publishing Ltd.We consider the processes of plastic flow localization in dipolar materials undergoing high speed shear deformations. The mathematical model of the processes of plastic flow localization is formulated taking into account dipolar effect. We introduce the numerical algorithm which is based on adaptive mesh refinement technique. We show that this algorithm allows to increase performance of computations. We also studied the statistical properties of shear bands formation. We show that dipolar effect changes average characteristics of the processes considered such as average temperature, stress and etc. Moreover this effect leads to increase in initiation time, changes the widths of localization zones and distances between them.
- ПубликацияОткрытый доступThe first integrals and exact solutions of a two-component Belousov-Zhabotinskii reaction system(2019) Kudryashov, N. A.; Кудряшов, Николай Алексеевич© 2019 Published under licence by IOP Publishing Ltd.The popular Belousov-Zhabotinskii (BZ) system of equations for description of a two component reaction is considered. The Painlevé test is applied to determine integrability of this system. It is shown that the system of equations is nonintegrable in general case. The parameter values of the mathematical model are found for the case when the system of equations passes the Painlevé test. Simplest solutions of the system of equations are presented. Additional conditions are specified when the general solutions of the system can be found. These general solutions are found using the new generalized method for finding exact solutions and the first integrals. Two first integrals of the Belousov-Zhabotinskii reaction system are given at additional conditions on parameter values for the mathematical model.
- ПубликацияОткрытый доступNonlinear differential equations associated with the first Painleve hierarchy(2019) Kudryashov, N. A.; Кудряшов, Николай Алексеевич© 2018 Elsevier Ltd The first Painlevé hierarchy with general solutions in the form of the Painlevé transcendents is considered. The linear system associated with this hierarchy is given. Some new hierarchies with properties similar to the Painlevé hierarchies are presented. It is shown that the solutions of these hierarchies are expressed via the transcendents of the first Painlevé hierarchy. Thus, the list of nonlinear differential equations whose solutions are expressed in terms of non-classical functions is extended.
- ПубликацияОткрытый доступAnalytical features of the SIR model and their applications to COVID-19(2021) Vigdorowitsch, M.; Kudryashov, N. A.; Chmykhov, M. A.; Кудряшов, Николай Алексеевич; Чмыхов, Михаил Александрович© 2020 Elsevier Inc.A classic two-parameter epidemiological SIR-model of the coronavirus propagation is considered. The first integrals of the system of non-linear equations are obtained. The Painlevé test shows that the system of equations is not integrable in the general case. However, the general solution is obtained in quadrature as an inverse time-function. Using the first integrals of the system of equations, analytical dependencies for the number of infected patients I(t) and that of recovered patients R(t) on the number of susceptible to infection S(t) are obtained. A particular attention is paid to interrelation of I(t) and R(t) both depending on α/β, where α is the contact rate in the community and β is the intensity of recovery/decease of patients. It is demonstrated that the data on particular morbidity waves in Hubei (China), Italy, Austria, South Korea, Moscow (Russia) as well some Australian territories are satisfactorily described by the expressions obtained for I(R). The variability of parameter N having been traditionally considered as a static population size is discussed.