Theory Of Nonlinear Oscillations Research Articles

Evolution of a stellar system in the context of the virial equation

The virial equation is used to clarify the nature of the dynamic evolution of a stellar system. The methods used are based on analytical and numerical modeling of evolution, as well as on an approach long used in the nonlinear theory of oscillations. It is shown that the mean harmonic radius of a system with negative total energy never exceeds two times the equilibrium value. The time to reach the virial equlibrium state Tv is about two to three dozen dynamic time periods Td. For systems not in close proximity to virial equilibrium, the virial ratio, the mean harmonic radius, and the root mean square radius of the system fluctuate during Tv; then the virial ratio and mean harmonic radius stabilize near their equilibrium values, while the root mean square radius continues to increase (possibly ad infinitum). Thus, the moment of inertia of the system relative to the center of gravity and its potential energy have significantly different behavior, which leads to the formation of a relatively small quasi-equilibrium core and an extended halo.

BIFURCATION OF TORI FOR PARABOLIC SYSTEMS OF DIFFERENTIAL EQUATIONS WITH WEAK DIFFUSION

The aim of the present article is to investigate of some properties of quasiperiodic solutions of nonlinear autonomous parabolic systems with the periodic condition. The research is devoted to the investigation of parabolic systems of differential equations with the help of integral manifolds method in the theory of nonlinear oscillations. We prove the existence of quasiperiodic solutions in autonomous parabolic system of differential equations with weak diffusion on the circle. We study existence and stability of an arbitrarily large finite number of tori for a parabolic system with weak diffusion. The quasiperiodic solution of parabolic system is sought in the form of traveling wave. A representation of the integral manifold is obtained. We seek a solution of parabolic system with the periodic condition in the form of a Fourier series in the complex form and introduce the norm in the space of the coefficients in the Fourier expansion. We use the normal forms method in the general parabolic system of differential equations with weak diffusion. We use bifurcation theory for ordinary differential equations and quasilinear parabolic equations. The existence of quasiperiodic solutions in an autonomous parabolic system of differential equations on the circle with small diffusion is proved. The problems of existence and stability of traveling waves in the parabolic system with weak diffusion are investigated.

Existence and stability of traveling waves in parabolic systems of differential equations with weak diffusion

The aim of the present paper is to investigate of some properties of periodic solutions of a nonlinear autonomous parabolic systems with a periodic condition. We investigate parabolic systems of differential equations using an integral manifolds method of the theory of nonlinear oscillations. We prove the existence of periodic solutions in an autonomous parabolic system of differential equations with weak diffusion on the circle. We study the existence and stability of an arbitrarily large finite number of cycles for a parabolic system with weak diffusion. The periodic solution of parabolic equation is sought in the form of traveling wave. A representation of the integral manifold is obtained. We seek a solution of parabolic system with the periodic condition in the form of a Fourier series in the complex form and introduce a norm in the space of the coefficients in the Fourier expansion. We use the normal forms method in the general parabolic system of differential equations with retarded argument and weak diffusion. We use bifurcation theory for delay differential equations and quasilinear parabolic equations. The existence of periodic solutions in an autonomous parabolic system of differential equations on the circle with retarded argument and small diffusion is proved. The problems of existence and stability of traveling waves in the parabolic system with retarded argument and weak diffusion are investigated.

To Integration of the Damped Mathieu Equation in the Monograph of N. N. Bogoliubov and Y. A. Mitropolsky “Asymptotic Methods in the Theory of Nonlinear Oscillations”

To Integration of the Damped Mathieu Equation in the Monograph of N. N. Bogoliubov and Y. A. Mitropolsky “Asymptotic Methods in the Theory of Nonlinear Oscillations”

The boundary value problems for the linear singularly perturbed systems of differential-algebraic equations

The article considers the main stages of the development of the theory of asymptotic integration of boundary-value problems for linear singularly perturbed differential-algebraic systems. The need of developing constructive methods of approximate integration of boundary-value problems for differential-algebraic systems is due to the importance of their practical application in the theory of nonlinear oscillations, stability of motion, control theory, radio engineering, and biology.
 In the present paper the authors offer a review of literary sources, which consider the methods of constructing asymptotic solutions of singularly perturbed systems of differential equations with a degenerate matrix with derivatives under the condition of stability of the spectrum of the limit pencil of matrices. It is noted that the problem of constructing asymptotic solutions of boundary-value problems for systems of this type is poorly studied, and therefore relevant. In particular, the question of the conditions for the existence and uniqueness of the solutions of these problems and the development of methods for constructing their asymptotics in various cases related to the behavior of the spectrum of the limit pencil of matrices has been poorly researched.

Linear differential-algebraic boundary value problem with singular pulse influence

The study of differential-algebraic boundary value problems was initiated in the works of K. Weierstrass, N. N. Luzin and F. R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the work of S. Campbell, Yu. E. Boyarintsev, V. F. Chistyakov, A. M. Samoilenko, M. O. Perestyuk, V. P. Yakovets, O. A. Boichuk, A. Ilchmann and T. Reis. The study of the differential-algebraic boundary value problems is associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, theory of control, theory of motion stability. At the same time, the study of differential algebraic boundary value problems is closely related to the study of pulse boundary value problems for differential equations, initiated M. O. Bogolybov, A. D. Myshkis, A. M. Samoilenko, M. O. Perestyk and O. A. Boichuk. Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A. M. Samoilenko, M. O. Perestyuk and O. A. Boichuk on a pulse linear boundary value problems for differential-algebraic equations, in particular finding the necessary and sufficient conditions for the existence of the desired solutions, and also the construction of the Green’s operator of the Cauchy problem and the generalized Green operator of a pulse linear boundary value problem for a differential-algebraic equation. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The proposed scheme of the research of the linear differential-algebraic boundary value problem for a differential-algebraic equation with impulse action in the critical case in this article can be transferred to the linear differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The above scheme of the analysis of the seminonlinear differential-algebraic boundary value problems with impulse action generalizes the results of S. Campbell, A. M. Samoilenko, M. O. Perestyuk and O. A. Boichuk and can be used for proving the solvability and constructing solutions of weakly nonlinear boundary value problems with singular impulse action in the critical and noncritical cases.

Linear differential-algebraic boundary value problem with nonsingular pulse influence

The study of differential-algebraic boundary value problems was initiated in the works of K. Weierstrass, N.N. Luzin and F.R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the work of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boichuk, A. Ilchmann and T. Reis. The study of the differential-algebraic boundary value problems is associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, theory of control, theory of motion stability. At the same time, the study of differential algebraic boundary value problems is closely related to the study of pulse boundary value problems for differential equations, initiated M.O. Bogolybov, A.D. Myshkis, A.M. Samoilenko, M.O. Perestyk and O.A. Boichuk. Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A.M. Samoilenko, M.O. Perestyuk and O.A Boichuk on a pulse linear boundary value problems for differential-algebraic equations, in particular finding the necessary and sufficient conditions for the existence of the desired solutions, and also the construction of the Green's operator of the Cauchy problem and the generalized Green operator of a pulse linear boundary value problem for a differential-algebraic equation. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a differential-algebraic equation with nonsingular impulse action. The proposed scheme of the research of the linear differential-algebraic boundary value problem for a differential-algebraic equation with impulse action in the critical case in this article can be transferred to the linear differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The above scheme of the analysis of the seminonlinear differential-algebraic boundary value problems with impulse action neralizes the results of S. Campbell, A.M. Samoilenko, M.O. Perestyuk and O.A Boichuk and can be used for proving the solvability and constructing solutions of weakly nonlinear boundary value problems with singular impulse action in the critical and noncritical cases.

The Class of Self-Oscillating Systems with Analytically Defined Limiting Cycles

It is studied the model that generalizes the well-known equations of the nonlinear oscillation theory (Van der Pol and Rayleigh). The limit cycles of the model are determined by the total energy of the oscillation in the absence of dissipation/energy input into the system. In contrast to the traditional equations of self-oscillations, equations in the Lagrange form are used here. Phase portraits of limit cycles which can be used in engineering are constructed analytically and numerically.

ON THE EXISTENCE OF A CONDITIONALLY PERIODIC SOLUTION OF A QUASILINEAR SYSTEM DIFFERENTIAL EQUATION IN THE CRITICAL CASE

In the theory of nonlinear oscillations one often encounters conditionally periodic oscillations resulting from the superposition of several oscillations with frequencies incommensurable with each other. When finding a solution to a resonant quasilinear differential system in the form of a conditionally periodic function, the problem of a small denominator arises. Consequently, the proof of the existence and even more the construction of such a solution is not an easy task. In this article, drawing on the work of V.I. Arnold, I. Moser, and other researchers proved the existence and constructed a conditionally periodic solution of a second-order quasilinear differential system in the critical case. Accelerated convergence method by N.N. Bogolyubova, Yu.A. Mitropolsky, A.M. Samoylenko. The result can be applied to construct a conditionally periodic solution of specific differential systems.

Evaluation of the pile interaction at vertical vibrations of foundation

The aim of the work - to research changes in the dynamic stiffness of pile foundations from the distance between the piles during vertical vibrations using experimental data from literature sources and solving wave models describing the vertical vibrations of a thin plate with round cuts. Methods. To verify the reliability of solutions of wave models describing changes in dynamic stiffness during vertical vibrations of pile foundations, it is used data obtained in experiments to determine the eigenfrequencies of 33 groups of friction piles with different distances between them. Also the data obtained from forced vertical oscillations of 22 groups of friction piles connected by pile-caps at different loads and distances between piles was used. In the processing of available amplitude-frequency curves for determining the dynamic stiffness the inverse problem is solved using the theory of nonlinear oscillations. The correspondence between the measured and predicted data is evaluated, when describing the behavior of the pile-ground system. Results. It is established that the relations that take into account the mutual influence of piles in the group, obtained in the framework of wave models solutions and used for calculating dynamic stiffness in vertical vibrations of pile foundations, allow us to obtain satisfactory results in accuracy. The deviation of the calculation results from the experimental data does not exceed 15%.

Investigation of solutions of weakly perturbed boundary value problems for systems with impulse action

The paper is devoted to the investigation of the existence and construction of solutions of weakly perturbed linear boundary value problems for systems with impulse action in the case when the generating boundary value problem with impulse action has no solutions for arbitrary right-hand sides. The relevance of this topic is due primarily to the importance of practical application of the theory of boundary value problems in various fields of science and technology - the theory of nonlinear oscillations, the theory of motion stability, control theory, a variety of geophysical problems. On the other hand, the question of the existence and construction of solutions of boundary value problems occupies one of the central and fundamentally important places in the qualitative theory of ordinary differential equations.

Essay of history of topological education in Nizhny Novgorod

The history of the development of topological education in Nizhny Novgorod is described — from the first lecture on topology for schoolchildren, given in 1939 by Professor A.G. Mayer, to the present. The need for knowledge of topology for further research was first understood in Nizhny Novgorod by representatives of the school of academician A.A. Andronov on the theory of nonlinear oscillations and the qualitative theory of differential equations. An important moment was organization by S.I. Alber in 1964 Gorky topological school, in which many outstanding mathematicians took part (D.V. Anosov, M.L. Gromov, S.P. Novikov, Y.G. Sinai, etc.) However, the specialist in real algebraic geometry, Professor D.A. Gudkov, became the motor for introducing topology into the educational process. This activity took place in close cooperation with Leningrad Professor V.A. Rokhlin and his students O.Ya. Viro and V.M. Kharlamov

Class of Auto-Oscillating Systems

This article proposes a model that generalizes the well-known equations of the nonlinear theory of oscillations (Van der Pol and Rayleigh equations), the limit cycles of which are the curves of the phase plane, determined by the total energy of oscillations in the absence of dissipation/energy inflow into the system. By changing the parameters of the system and the type of force action, it is possible to set the oscillation characteristics that are resistant to disturbances. The phase portraits of the system are considered, including the case of multiply connected limit cycles.

Nonlinear radiation oscillator theory for symmetric and anti-symmetric damping graphene coupled metamaterials

We develop a novel nonlinear radiation oscillator theory, which can describe nonlinear optical responses in symmetric and anti-symmetric damping graphene coupled metamaterials. An ultra-high quality factor of induced transparency can be realized in the proposed model. The results show that the spectral response and quality factor can be effectively tuned by the resonant detuning, the phase difference, the damping factor as well as the field intensity of the pump light. Moreover, the quality factor of induced transparency is increased by more than 60 times when the field intensity of the pump light and the damping factor are changed. Especially, the single induced transparency window splits into double induced transparency windows as the damping factors of dark mode and bright mode show anti-symmetric damping case. These results may pave the way for designing the high performance of nano-optical devices in terahertz.

On a non-linear droplet oscillation theory via the unified method

Presently, the oscillation of a liquid droplet in a dynamically negligible outer medium subject to surface tension and small viscosity is investigated. By using the potential flow assumption, the unified transform method by Fokas is employed to reduce the corresponding free boundary problem formulated on a time-dependent domain into a nonlinear system of integro-differential equations (IDEs). This new system depends on one less spatial variable and is now defined on a time-independent domain. Most importantly, the resulting set of equations governs the general droplet oscillation with arbitrarily large deviations from the spherical shape. As the nonlinearity of the above IDE system up to now prevented an analytical solution, the Poincaré expansion technique is employed, retaining terms up to the second order. By decomposing the unknowns into normal modes, these equations are uncoupled and the resulting ordinary differential equations for the mode amplitudes are solved, and the results are compared to those of previous works. It should be stressed that the present analysis is limited to small viscosity, or, in other words, for small Ohnesorge numbers. The reason for this is that, inside of the droplet, a potential flow is assumed and the viscous effect is taken into account only at the droplet surface by the jump condition of momentum. This is only reasonable for a small viscosity and a short time. Otherwise, vorticity is generated at the interface and diffuses toward the inside of the droplet.

NONSINGULAR INTEGRO-DIFFERENTIAL BOUNDARY VALUE PROBLEM NOT SOLVED WITH RESPECT TO THE DERIVATIVE

The study of the differential-algebraic boundary value problems was established in the papers of K. Weierstrass, M.M. Lusin and F.R. Gantmacher. Works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boi- chuk, A. Ilchmann and T. Reis are devoted to the systematic study of differential-algebraic boundary value problems. At the same time, the study of differential-algebraic boundary-value problems is closely related to the study of linear boundary-value problems for ordinary di- fferential equations, initiated in the works of A. Poincare, A.M. Lyapunov, M.M. Krylov, N.N. Bogolyubov, I.G. Malkin, A.D. Myshkis, E.A. Grebenikov, Yu.A. Ryabov, Yu.A. Mitropolsky, I.T. Kiguradze, A.M. Samoilenko, M.O. Perestyuk and O.A. Boichuk. The study of the linear differential-algebraic boundary value problems is connected with numerous applications of corresponding mathematical models in the theory of nonlinear osci- llations, mechanics, biology, radio engineering, the theory of the motion stability. Thus, the actual problem is the transfer of the results obtained in the articles and monographs of S. Campbell, A.M. Samoilenko and O.A. Boichuk on the linear boundary value problems for the integro-differential boundary value problem not solved with respect to the derivative, in parti- cular, finding the necessary and sufficient conditions of the existence of the desired solutions of the linear integro-differential boundary value problem not solved with respect to the derivative. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian integro-differential boundary value problem not solved with respect to the derivative. The proposed scheme of the research of the nonlinear Noetherian integro-differential boundary value problem not solved with respect to the derivative in the critical case in this article can be transferred to the seminonlinear integro-differential boundary value problem not solved with respect to the derivative.

Linear Noetherian boundary value problem for a matrix difference-algebraic Lyapunov equation

The study of differential-algebraic boundary value problems was initiated in the works of K. Weierstrass, N. N. Luzin and F. R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the work of S. Campbell, Yu. E. Boyarintsev, V. F. Chistyakov, A. M. Samoilenko, M. O. Perestyuk, V. P. Yakovets, O. A. Boichuk, A. Ilchmann and T. Reis. The study of the differential-algebraic boundary value problems is associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, theory of control, theory of motion stability. At the same time, the study of differential algebraic boundary value problems is closely related to the study of boundary value problems for difference equations, initiated in A. A. Markov, S. N. Bernstein, Ya. S. Besikovich, A. O. Gelfond, S. L. Sobolev, V. S. Ryaben’kii, V. B. Demidovich, A. Halanay, G. I. Marchuk, A. A. Samarskii, Yu. A. Mitropolsky, D. I. Martynyuk, G. M. Vayniko, A. M. Samoilenko, O. A. Boichuk and O. M. Standzhitsky. Study of nonlinear singularly perturbed boundary value problems for difference equations in partial differences is devoted to the work of V. P. Anosov, L. S. Frank, P. E. Sobolevskii, A. L. Skubachevskii and A. Asheraliev. Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A. M. Samoilenko and O. A. Boichuk on linear boundary value problems for difference-algebraic equations, in particular finding the necessary and sufficient conditions for the existence of the desired solutions, and also the construction of the Green’s operator of the Cauchy problem and the generalized Green operator of a linear boundary value problem for a difference-algebraic equation. Thus, the actual problem is the transfer of the results obtained in the articles and monographs of S. Campbell, A. M. Samoilenko and O. A. Boichuk on the linear boundary value problems for the differential-algebraic boundary value problem for a matrix Lyapunov equation, in particular, finding the necessary and sufficient conditions of the existence of the desired solutions of the linear differential-algebraic boundary value problem for a matrix Lyapunov equation. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a matrix Lyapunov equation. The proposed scheme of the research of the linear differential-algebraic boundary value problem for a matrix Lyapunov equation in the critical case in this article can be transferred to the seminonlinear differential-algebraic boundary value problem for a matrix Lyapunov equation.

On the solvability of the degenerate Noetherian difference-algebraic boundary value problem

The original conditions of solvability and the scheme of finding solutions of a linear Noetherian difference-algebraic boundary-value problem are proposed in the article, while the technique of pseudoinversion of matrices by Moore-Penrose is substantially used. The problem posed in the article continues to study the conditions for solvability of linear Noetherian boundary value problems given in the monographs of A.M. Samoilenko, A.V. Azbelev, V.P. Maximov, L.F. Rakhmatullina and A.A. Boichuk. The study of differential-algebraic boundary-value problems is closely related to the investigation of boundary-value problems for difference equations, initiated in the works of A.A. Markov, S.N. Bernstein, Y.S. Bezikovych, O.O. Gelfond, S.L. Sobolev, V.S. Ryabenkyi, V.B. Demidovych, A. Halanai, G.I. Marchuk, A.A. Samarskyi, Yu.A. Mytropolskyi, D.I. Martyniuk, G.M. Vainiko, A.M. Samoilenko and A.A. Boichuk. On the other hand, the study of boundary-value problems for difference equations is related to the study of differential-algebraic boundary-value problems initiated in the papers of K. Weierstrass, N.N. Lusin and F.R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, N.A. Perestiyk, V.P. Yakovets, A.A. Boichuk, A. Ilchmann and T. Reis. The study of differential-algebraic boundary value problems is also associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, control theory, motion stability theory. The general case of a linear bounded operator corresponding to the homogeneous part of a linear Noetherian difference-algebraic boundary value problem has no inverse is investigated. The generalized Green operator of a linear difference-algebraic boundary value problem is constructed in the article. The relevance of the study of solvability conditions, as well as finding solutions of linear Noetherian difference-algebraic boundary-value problems, is associated with the widespread use of difference-algebraic boundary-value problems obtained by linearizing nonlinear Noetherian boundary-value problems for systems of ordinary differential and difference equations. Solvability conditions are proposed, as well as the scheme of finding solutions of linear Noetherian difference-algebraic boundary value problems are illustrated in detail in the examples.

On discrete groups and solvable equations of nonlinear oscillations

The autonomous ordinary differential equation of Lienard is considered. This equation and its generalizations arise in the theory of nonlinear oscillations, dynamical systems, etc. Currently for equations of the Lienard type, the problem of finding exact solutions is relevant. In view of this, the problem is to find connections in Lienard type equations with investigated nonlinear equations. These problems are solved in this article for the Lienard equation with three-parametrical coefficients of polynomial type. We found a transformation that reduces the equation under consideration of the well-studied generalized Emden-Fowler equation. We indicated the method of finding exact solutions of Lienard equations with various parameters. The transformations discrete group of dihedron for the class of generalized Emden-Fauler equations induces a discrete group of transformations for the corresponding class of Lienard equations. Likewise, known solutions of some particular generalized Emden-Fauler equations induce solutions of the Lienard equations. As examples, we found exact solutions in elementary (including polynomials) and special functions of several Lienard equations with certain parameters.

A Disturbance Rejection Method for Beam Balance Oscillation Period Measurement

Precision measurement of the oscillation period of a beam balance oscillator has a fundamental role in the inertial mass measurement project. In order to improve the accuracy of oscillation period measurement, the nonlinear oscillation theory is used to study the periodic oscillation process of the beam balance oscillator in this paper, and the analytical solutions of the dynamic equations of the balance under free and forced conditions are deduced. Moreover, the effects of ground vibration, air turbulence, and temperature fluctuation on the accuracy of oscillation period measurement are analyzed in detail in both vacuum and atmospheric environments, and a method based on beam structural parameters adjustment and temperature coefficient correction is proposed to suppress these disturbances under air condition. Experimental results show that the relative standard uncertainty of the measured oscillation period can be reduced from $2.5\times 10^{-5}$ to $1.2\times 10^{-6}$ (Class A, $k =1$ ) after the above-mentioned method is adopted.