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Vol. 15, no. 3

Vol. 15, no. 3, 2019

Misyurin S. Y.,  Kreinin G. V.,  Nosova N. Y.
Abstract
Mathematical models describing technically oriented dynamical systems are generally rather complex. Very time-consuming interactive procedures have to be used when selecting the structure and parameters of the system. Direct enumeration of options using such procedures can be avoided by applying a number of means, in particular, dimension methods and similarity theory. The use of dimension and similarity theory along with the general qualitative analysis of the system can serve as an effective theoretical research method. At the same time, these theories are simple. Using dimension and similarity theory, it is possible to draw conclusions when considering phenomena that depend on a large number of parameters, but so that some of them become insignificant in certain cases.
The combined method of using the theory of similarity, analogousness and methods developed by the authors for testing the drive model provides insight into its dynamics, controllability and other properties. The proposed approach is based on systematization and optimization of the process of forming a dimensionless model and similarity criteria, its focus on solving the formulated problem, as well as on special methods of modeling and processing of simulation results. It improves the efficiency of using similarity properties in solving analysis and synthesis problems. The advantage of this approach manifests itself in the ultimate simplification of the dimensionless model compared to the original model. The reduced (dimensionless) model is characterized by a high versatility and efficiency of finding the optimal and final solution in the selection of parameters of the real device, as it contains a significantly smaller number of parameters, which makes it convenient in solving problems of analysis and, in particular, synthesis of the system.
Dimension methods and similarity theory are successfully applied in the study of dynamical systems of different classes. The problems that arise are mainly related to the selection of a rational combination of the main units of measurement of physical quantities, the transition to dimensionless models and the formation of basic similarity criteria. The structure and the form of the dimensionless model depend on the adopted units of measurement of the variables appearing in the equations of the model and on the expressions assigned to its coefficients. Specified problems are solved by researchers, as a rule, by appealing to their intuition and experience. Meanwhile, there exist well-known systematized approaches to solving similar problems based on the method of the theory of analogousness.
Keywords: similarity, analogousness, hydraulic drive, dynamical system, dimensionless parameters
Citation: Misyurin S. Y.,  Kreinin G. V.,  Nosova N. Y., Similarity and Analogousness in Dynamical Systems and Their Characteristic Features, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp. 213-220
DOI:10.20537/nd190301
Safronov A. A.,  Koroteev A. A.,  Filatov N. I.,  Safronova N. A.
Abstract
Stationary waves in a cylindrical jet of a viscous fluid are considered. It is shown that when the capillary pressure gradient of the term with the third derivative of the jet radius in the axial coordinate is taken into account in the expression, the previously described self-similar solutions of hydrodynamic equations arise. Solutions of the equation of stationary waves propagation are studied analytically. The form of stationary soliton-like solutions is calculated numerically. The results obtained are used to analyze the process of thinning and rupture of jets of viscous liquids.
Keywords: instability, capillary flows, viscous jet, stationary waves
Citation: Safronov A. A.,  Koroteev A. A.,  Filatov N. I.,  Safronova N. A., Capillary Hydraulic Jump in a Viscous Jet, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp. 221-231
DOI:10.20537/nd190302
Mogilevich L.,  Ivanov S.
Abstract
This article is devoted to studying longitudinal deformation waves in physically nonlinear elastic shells with a viscous incompressible fluid inside them. The impact of construction damping on deformation waves in longitudinal and normal directions in a shell, and in the presence of surrounding medium are considered.
The presence of a viscous incompressible fluid inside the shell and the impact of fluid movement inertia on the wave velocity and amplitude are taken into consideration. In the case of a shell filled with a viscous incompressible fluid, it is impossible to study deformation wave models by qualitative analysis methods. This makes it necessary to apply numerical methods. The numerical study of the constructed model is carried out by means of a difference scheme analogous to the Crank – Nickolson scheme for the heat conduction equation. The amplitude and velocity do not change in the absence of surrounding medium impact, construction damping in longitudinal and normal directions, as well as in the absence of fluid impact. The movement occurs in the negative direction, which means that the movement velocity is subsonic. The numerical experiment results coincide with the exact solution, therefore, the difference scheme and the modified Korteweg – de Vries – Burgers equation are adequate.
Keywords: nonlinear waves, elastic cylinder shell, viscous incompressible fluid, Crank – Nickolson difference scheme
Citation: Mogilevich L.,  Ivanov S., The Study of Wave Propagation in a Shell with Soft Nonlinearity and with a Viscous Liquid Inside, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp. 233-250
DOI:10.20537/nd190303
Reutov V.,  Rybushkina G. V.
Abstract
The anomalous transport of a passive scalar at the excitation of immovable chains of wave structures with closed streamlines in a barotropic reverse jet flow is studied. The analysis is performed for a plane-parallel flow in a channel between rigid walls in the presence of the beta effect and external friction. Periodic boundary conditions are set along the channel, while nonpercolation and sticking conditions are adopted on the channel walls. The equations of a barotropic (quasi-two-dimensional) flow are solved numerically using a pseudospectral method. A reverse jet with a “two-hump” asymmetric velocity profile facilitating the faster transition to the complex dynamics of the Eulerian flow fields is considered. Unlike the most developed kinematic models of anomalous transport, the basic chain of structures becomes unsteady due to the birth of supplementary perturbations at saturation of barotropic instability. A regular (multiharmonic) regime of wave generation is shown to appear due to the excitation of a new flow mode. Immovable structure chains giving rise to anomalous transport are obtained in the multiharmonic and chaotic regimes. The velocity of the chains of structures was determined by watching movies made according to the computations of the streamlines. It is revealed that the onset of anomalous transport in a regular regime is possible at essentially lower supercriticality compared to the chaotic regime. Trajectories of the tracer particles containing alternations of long flights and oscillations are drawn in the chaotic regime. The time dependences of the averaged (over ensemble) displacement of the tracer particles and its variance are obtained for two basic regimes of generation with immovable chains of structures, and the corresponding exponents of the power laws are determined. Normal advection is revealed in the regular regime, while anomalous diffusion arises in both regimes and may be classified as a “superdiffusion”.
Keywords: barotropic reverse jet flow, chains of wave structures, dynamical chaos, anomalous advection and diffusion
Citation: Reutov V.,  Rybushkina G. V., Dynamical Model for the Anomalous Transport of a Passive Scalar in a Reverse Barotropic Jet Flow, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp. 251-260
DOI:10.20537/nd190304
Shvarts K. G.
Abstract
This paper presents a derivation of new exact solutions to the Navier – Stokes equations in Boussinesq approximation describing two advective flows in a rotating thin horizontal fluid layer with no-slip or free boundaries in a vibrational field. The layer rotates at a constant angular velocity; the axis of rotation is aligned with the vertical axis of coordinates. The temperature is linear along the boundaries of the layer. The case of longitudinal vibration is considered. The resulting solutions are similar to those describing the advective flows in a rotating fluid layer with solid or free boundaries without vibration. In both cases, the velocity profile is antisymmetric. Thus, in particular, in the absence of rotation, the longitudinal vibration in the presence of advection can be considered as a kind of “one-dimensional” rotation. The presence of rotation initiates the vortex motion of the fluid in the layer. Longitudinal vibration has a stronger effect on the xth component of the velocity than on the yth component. At large values of the Taylor number and (or) the vibration analogue of the Rayleigh number thin boundary layers of velocity, temperature and amplitude of the pulsating velocity component arise, the thickness of which is proportional to the root of the fourth degree from the sum of these numbers.
Keywords: horizontal convection, longitudinal vibration, exact solution
Citation: Shvarts K. G., Advective Flow of a Rotating Fluid Layer in a Vibrational Field, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp. 261-270
DOI:10.20537/nd190305
Privalova V.,  Prosviryakov E. Y.,  Simonov M. A.
Abstract
A new exact solution to the Navier – Stokes equations is obtained. This solution describes the inhomogeneous isothermal Poiseuille flow of a viscous incompressible fluid in a horizontal infinite layer. In this exact solution of the Navier – Stokes equations, the velocity and pressure fields are the linear forms of two horizontal (longitudinal) coordinates with coefficients depending on the third (transverse) coordinate. The proposed exact solution is two-dimensional in terms of velocity and coordinates. It is shown that, by rotation transformation, it can be reduced to a solution describing a three-dimensional flow in terms of coordinates and a two-dimensional flow in terms of velocities. The general solution for homogeneous velocity components is polynomials of the second and fifth degrees. Spatial acceleration is a linear function. To solve the boundaryvalue problem, the no-slip condition is specified on the lower solid boundary of the horizontal fluid layer, tangential stresses and constant horizontal (longitudinal) pressure gradients specified on the upper free boundary. It is demonstrated that, for a particular exact solution, up to three points can exist in the fluid layer at which the longitudinal velocity components change direction. It indicates the existence of counterflow zones. The conditions for the existence of the zero points of the velocity components both inside the fluid layer and on its surface under nonzero tangential stresses are written. The results are illustrated by the corresponding figures of the velocity component profiles and streamlines for different numbers of stagnation points. The possibility of the existence of zero points of the specific kinetic energy function is shown. The vorticity vector and tangential stresses arising during the flow of a viscous incompressible fluid layer under given boundary conditions are analyzed. It is shown that the horizontal components of the vorticity vector in the fluid layer can change their sign up to three times. Besides, tangential stresses may change from tensile to compressive, and vice versa. Thus, the above exact solution of the Navier – Stokes equations forms a new mechanism of momentum transfer in a fluid and illustrates the occurrence of vorticity in the horizontal and vertical directions in a nonrotating fluid. The three-component twist vector is induced by an inhomogeneous velocity field at the boundaries of the fluid layer.
Keywords: Poiseuille flow, gradient flow, exact solution, counterflow, stagnation point, vorticity
Citation: Privalova V.,  Prosviryakov E. Y.,  Simonov M. A., Nonlinear Gradient Flow of a Vertical Vortex Fluid in a Thin Layer, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp. 271-283
DOI:10.20537/nd190306
Hussein A.
Abstract
We consider a quit general problem of motion of an asymmetric rigid body about a fixed point, acted upon by an irreducible skew combination of gravitational, electric and magnetic fields. Two of those three fields are uniform and the third has a more complicated structure. The existence of precessional motions about a nonvertical axis is established. Conditions on the parameters of the system are obtained. An alternative physical interpretation is given in the framework of the problem of motion of a rigid body immersed in an incompressible perfect fluid, acted upon by torques due to two uniform fields.
Keywords: rigid body, precessional motion, three irreducible fields
Citation: Hussein A., Precessional Motion of a Rigid Body Acted upon by Three Irreducible Fields, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp. 285-292
DOI:10.20537/nd190307
Mukherjee I.,  Guha P.
Abstract
The nonholonomic deformations of nonlocal integrable systems belonging to the nonlinear Schrödinger family are studied using the bi-Hamiltonian formalism as well as the Lax pair method. The nonlocal equations are first obtained by symmetry reductions of the variables in the corresponding local systems. The bi-Hamiltonian structures of these equations are explicitly derived. The bi-Hamiltonian structures are used to obtain the nonholonomic deformation following the Kupershmidt ansatz. Further, the same deformation is studied using the Lax pair approach and several properties of the deformation are discussed. The process is carried out for coupled nonlocal nonlinear Schrödinger and derivative nonlinear Schrödinger (Kaup Newell) equations. In the case of the former, an exact equivalence between the deformations obtained through the bi-Hamiltonian and Lax pair formalisms is indicated.
Keywords: nonlocal integrable systems, nonlinear Schr¨odinger equation, Kaup –Newell equation, bi-Hamiltonian system, Lax method, nonholonomic deformation
Citation: Mukherjee I.,  Guha P., A Study of Nonholonomic Deformations of Nonlocal Integrable Systems Belonging to the Nonlinear Schrödinger Family, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp. 293-307
DOI:10.20537/nd190308
Grillo S. D.,  Salomone L. M.,  Zuccalli M.
Abstract
For an underactuated (simple) Hamiltonian system with two degrees of freedom and one degree of underactuation, a rather general condition that ensures its stabilizability, by means of the existence of a (simple) Lyapunov function, was found in a recent paper by D.E. Chang within the context of the energy shaping method. Also, in the same paper, some additional assumptions were presented in order to ensure also asymptotic stabilizability. In this paper we extend these results by showing that the above-mentioned condition is not only sufficient, but also necessary. And, more importantly, we show that no additional assumption is needed to ensure asymptotic stabilizability.
Keywords: underactuated systems, Hamiltonian systems, asymptotic stability, Lyapunov functions
Citation: Grillo S. D.,  Salomone L. M.,  Zuccalli M., Asymptotic Stabilizability of Underactuated Hamiltonian Systems With Two Degrees of Freedom, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp. 309-326
DOI:10.20537/nd190309
Gorr G. V.,  Tkachenko D.,  Shchetinina E. K.
Abstract
The problem of the motion of a rigid body with a fixed point in a potential force field is considered. A new case of three nonlinear invariant relations of the equations of motion is presented. The properties of Euler angles, Rodrigues – Hamilton parameters, and angular velocity hodographs in the Poinsot method are investigated using an integrated approach in the interpretation of body motion.
Keywords: potential force field, Euler angles, Rodrigues – Hamilton parameters, Poinsot method
Citation: Gorr G. V.,  Tkachenko D.,  Shchetinina E. K., Research on the Motion of a Body in a Potential Force Field in the Case of Three Invariant Relations, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp. 327-342
DOI:10.20537/nd190310
Zubelevich O. E.
Abstract
This paper is concerned with the motion of a particle on a horizontal vibrating plane with Coulomb friction. It is proved that, when some constant force is added, the system has a periodic solution.
Keywords: classical mechanics, systems with friction, Filippov’s systems, periodic solutions, differential inclusions
Citation: Zubelevich O. E., A Particle on a Moving Plane with Coulomb Friction, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp. 343-349
DOI:10.20537/nd190311
Vetchanin  E. V.,  Mikishanina E. A.
Abstract
The dynamics of a body with a fixed point, variable moments of inertia and internal rotors are considered. A stability analysis of permanent rotations and periodic solutions of the system is carried out. In some simplest cases the stability analysis is reduced to investigating the stability of the zero solution of Hill’s equation. It is shown that by periodically changing the moments of inertia it is possible to stabilize unstable permanent rotations of the system. In addition, stable dynamical regimes can lose stability due to a parametric resonance. It is shown that, as the oscillation frequency of the moments of inertia increases, the dynamics of the system becomes close to an integrable one.
Keywords: Liouville equations, Euler –Poisson equations, Hill’s equation, Mathieu equation, parametric resonance, vibrostabilization, Euler – Poinsot case, Joukowski –Volterra case
Citation: Vetchanin  E. V.,  Mikishanina E. A., Vibrational Stability of Periodic Solutions of the Liouville Equations, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp. 351-363
DOI:10.20537/nd190312
Migalev A. S.,  Gotovtsev P. M.
Abstract
This paper addresses the spiking (or pulsed) neural network model with synaptic time delays at dendrites. This model allows one to change the action potential generation time more precisely with the same input activity pattern. The action potential time control principle proposed previously by several researchers has been implemented in the model considered. In the neuron model the required excitatory and inhibitory presynaptic potentials are formed by weight coefficients with synaptic delays. Various neural network architectures with a long-term plasticity model are investigated. The applicability of the spike-timing-dependent plasticity based learning rule (STDP) to a neuron model with synaptic delays is considered for a more accurate positioning of action potential time. Several learning protocols with a reinforcement signal and induced activity using varieties of functions of weight change (bipolar STDP and Ricker wavelet) are used. Modeling of a single-layer neural network with the reinforcement signal modulating the weight change function amplitude has shown a limited range of available output activity. This limitation can be bypassed using the induced activity of the output neuron layer during learning. This modification of the learning protocol allows reproducing more complex output activity, including for multiple layered networks. The ability to construct desired activity on the network output on the basis of a multichannel input activity pattern was tested on single and multiple layered networks. Induced activity during learning for networks with feedback connections allows one to synchronize multichannel input spike trains with required network output. The application of the weight change function leads to association of input and output activity by the network. When the induced activity is turned off, this association, configuration on the required output, remains. Increasing the number of layers and reducing feedback connection leads to weakening of this effect, so that additional mechanisms are required to synchronize the whole network.
Keywords: pulsed neural network model, spiking neural network model, synaptic plasticity, synchronization, induced activity, time delayed synapses
Citation: Migalev A. S.,  Gotovtsev P. M., Modeling the Learning of a Spiking Neural Network with Synaptic Delays, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp. 365-380
DOI:10.20537/nd190313
Kiselev O. M.
Abstract
We study the asymptotic behavior of nonlinear oscillators under an external driver with slowly changing frequency and amplitude. As a result, we obtain formulas for properties of the amplitude and frequency of the driver when the autoresonant behavior of the nonlinear oscillator is observed. Also, we find the measure of autoresonant asymptotic behaviors for such a driven nonlinear oscillator.
Keywords: nonlinear oscillator, autoresonance, perturbations
Citation: Kiselev O. M., Conditions for Phase Locking and Dephasing of Autoresonant Pumping, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp. 381-394
DOI:10.20537/nd190314

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