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# Online First Publication is available now!

 Krasil'nikov P. S.,  Maiorov A. Y. Abstract We study a mechanical system with two degrees of freedom simulating the motion of rotor blades on an elastic bushing of a medium-sized helicopter. For small values of some problem parameters, the destabilizing effect due to small linear viscous friction forces has been studied earlier. Here we study the problem with arbitrary large friction forces for arbitrary values of the problem parameters. In the plane of parameters, the regions of asymptotic stability and instability are calculated. As a result, necessary and sufficient conditions for the existence of a destabilizing effect under the action of potential, follower forces and arbitrary friction forces have been obtained. It is shown that, if some critical friction coefficient $k_*$ tends to infinity, then there exists a Ziegler area with arbitrarily large dissipative forces. Keywords: Non-conservative mechanical system, large linear dissipative forces, follower force, Ziegler’s effect, Lyapunov’s stability Citation: Krasil'nikov P. S.,  Maiorov A. Y., On Ziegler Effect with Arbitrary Large Viscous Friction, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 2, pp. 161-170 DOI:10.20537/nd220201
 Chernyshov A. V.,  Chernyshova S. A. Abstract The phenomenon of “bifurcation memory”, which can be detected during the steering of river vessels, is considered by researchers with the questions of equilibrium point’s bifurcations. It was noticed that in some dynamic systems during saddle-node bifurcation, areas are formed on the phase plane (so-called “phase spot”), passing through which the phase velocity of the representative point decreases. A decrease in phase velocity (e. g. during the steering of river vessel) can cause navigation accidents during shallow waters navigation. In order to improve sailing safety, it is necessary to investigate topological features of phase spots under any possible environmental conditions and steering angle values. In this paper, we propose a new method that makes it easy to get information about the localization of regions with decreased phase velocity. We gained the results related to the topology of areas of different types of motion (accelerated or decelerated) of representative point. In addition, the paper presents the process of evolution of these regions, according to the change of steering angles, as well as before and after the bifurcation. The new method, offered in the article, is more accurate for determining the boundaries of the areas compared to the methods of Feigin and Chirkova. This will allow us to make more correct predictions of changes in the dynamics of the object. The practical implication of the suggested method is that by using it, we can get information about the location of areas with different types of motion of the representative point on the phase plane for different values of steering angle and environmental conditions. This information can be used in the control algorithm of the driving object, for example, in order to predict the decreasing of phase velocities. Keywords: bifurcation memory, bifurcations in dynamic systems, steering of vessels, Static- Dynamic plane, phase spots Citation: Chernyshov A. V.,  Chernyshova S. A., A Method of Investigating the Phenomenon of Bifurcation Memory in the Dynamics of River Vessels, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 2, pp. 171-181 DOI:10.20537/nd220202
 Surulere S. A.,  Shatalov M. Y.,  Ehigie J. O.,  Adeniji A. A.,  Fedotov I. A. Abstract The oscillatory motion in nonlinear nanolattices having different interatomic potential energy functions is investigated. Potential energies such as the classical Morse, Biswas – Hamann and modified Lennard – Jones potentials are considered as interaction potentials between atoms in one-dimensional nanolattices. Noteworthy phenomena are obtained with a nonlinear chain, for each of the potential functions considered. The generalized governing system of equations for the interaction potentials are formulated using the well-known Euler – Lagrange equation with Rayleigh’s modification. Linearized damping terms are introduced into the nonlinear chain. The nanochain has statistical attachments of 40 atoms, which are perturbed to analyze the resulting nonlinearities in the nanolattices. The range of initial points for the initial value problem (presented as second-order ordinary differential equations) largely varies, depending on the interaction potential. The nanolattices are broken at some initial point(s), with one atom falling off the slender chain or more than one atom falling off. The broken nanochain is characterized by an amplitude of vibration growing to infinity. In general, it is observed that the nonlinear effects in the interaction potentials cause growing amplitudes of vibration, accompanied by disruptions of the nanolattice or by the translation of chaotic motion into regular motion (after the introduction of linear damping). This study provides a computationally efficient approach for understanding atomic interactions in long nanostructural components from a theoretical perspective. Keywords: nonlinear interactions, 1D lattices, Euler – Lagrange, interatomic potentials, nanostructures Citation: Surulere S. A.,  Shatalov M. Y.,  Ehigie J. O.,  Adeniji A. A.,  Fedotov I. A., Nonlinear Interactions in Nanolattices Described by the Classical Morse, Biswas – Hamann and Modified Lennard – Jones Potentials, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 2, pp. 183-201 DOI:10.20537/nd220103
 González-Gaxiola O.,  León-Ramírez A.,  Chacón-Acosta G. Abstract Recently, motivated by the interest in the problems of nonlinear dynamics of cylindrical shells, A. I. Zemlyanukhin et al. (Nonlinear Dyn, 98, 185–194, 2019) established the so-called Schamel – Kawahara equation (SKE). The SKE generalizes the well-known nonlinear Schamel equation that arises in plasma physics problems, by adding the high-order dispersive terms from the Kawahara equation. This article presents families of new solutions to the Schamel – Kawahara model using the Kudryashov method. By performing the symbolic computation, we show that this method is a valuable and efficient mathematical tool for solving application problems modeled by nonlinear partial differential equations (NPDE). Keywords: Schamel – Kawahara equation, Kudryashov method, exact solutions, nonlinear PDE Citation: González-Gaxiola O.,  León-Ramírez A.,  Chacón-Acosta G., Application of the Kudryashov Method for Finding Exact Solutions of the Schamel – Kawahara Equation, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 2, pp. 203-215 DOI:10.20537/nd220204
 Ekomasov E. G.,  Nazarov V. N.,  Samsonov K. Y. Abstract Possibility of changing the dynamic parameters of localized breather and soliton waves for the sine-Gordon equation in the model with extended impurity, variable external force and dissipation was investigated using the autoresonance method. The model of ferromagnetic structure consisting of two wide identical layers separated by a thin layer with modified values of magnetic anisotropy parameter was taken as a basis. Frequency of external field is a linear function of time. The sine-Gordon equation (SGE) was solved numerically using the finite differences method with explicit scheme of integration. For certain values of the extended impurity parameters a magnetic inhomogeneity in the form of magnetic breather is formed when domain wall passes through it with constant velocity. The numerical simulation showed that using special variable force and small amplitude it is possible to resonantly increase the amplitude of breather. For each case of the impurity parameters values, there is a threshold value of the magnetic field amplitude leading to resonance. Geometric parameters of thin layer also have influence on the resonance effect — for decreasing layer width the breather amplitude grows more slowly. For large layer width the translation mode of breather oscillations is also excited. For certain parameters of extended impurity, a soliton can form. For a special type of variable field with frequency linearly dependent on time, soliton is switched to antisoliton and vice versa. Keywords: autoresonance, sine-Gordon equation, spatially modulated periodic potential, impurities, kink, breather, soliton Citation: Ekomasov E. G.,  Nazarov V. N.,  Samsonov K. Y., Changing the Dynamic Parameters of Localized Breather and Soliton Waves in the Sine-Gordon Model with Extended Impurity, External Force, and Decay in the Autoresonance Mode, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 2, pp. 217-229 DOI:10.20537/nd220205
 Hosseini K.,  Mirzazadeh M.,  Dehingia K.,  Das A.,  Salahshour S. Abstract In the present paper, the authors are interested in studying a famous nonlinear PDE referred to as the $(2 + 1)$-dimensional chiral Schrцdinger (2D-CS) equation with applications in mathematical physics. In this respect, the real and imaginary portions of the 2D-CS equation are firstly derived through a traveling wave transformation. Different wave structures of the 2D-CS equation, classified as bright and dark solitons, are then retrieved using the modified Kudryashov (MK) method and the symbolic computation package. In the end, the dynamics of soliton solutions is investigated formally by representing a series of 3D-plots. Keywords: $(2 + 1)$-dimensional chiral Schrödinger equation, traveling wave transformation, modified Kudryashov method, different wave structures Citation: Hosseini K.,  Mirzazadeh M.,  Dehingia K.,  Das A.,  Salahshour S., A Study of Different Wave Structures of the $(2 + 1)$-dimensional Chiral Schrödinger Equation, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 2, pp. 231-241 DOI:10.20537/nd220206
 Shamin A. Y. Abstract This paper addresses the problem of the motion of the Chaplygin sleigh, a rigid body moving with three points in contact with a horizontal plane. One of them is equipped with a knife edge along which there is no slipping. Special attention is given to the case where dry friction is present at one of the points of support without the knife edge. The equations of motion of the body are written, the normal reactions are calculated, and the behavior of the phase curves in the neighborhood of an equilibrium point, depending on the geometric and mass characteristics of the body, is investigated by the method of introducing a small parameter. Keywords: dry friction, Chaplygin sleigh Citation: Shamin A. Y., On the Motion of the Chaplygin Sleigh Along a Horizontal Plane with Friction in the Asymmetric Case, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 2, pp. 243-251 DOI:10.20537/nd220207
 Dzhalilov A.,  Mayer D.,  Aliyev A. Abstract Let $T\in C^{2+\varepsilon}(S^{1}\setminus\{x_b^{}\})$, $\varepsilon>0$, be an orientation preserving circle homeomorphism with rotation number $\rho_T^{}=[k_1^{},\,k_2^{},\,\ldots,\,k_m^{},\,1,\,1,\,\ldots]$, $m\ge1$, and a single break point $x_b^{}$. Stochastic perturbations $\overline{z}_{n+1}^{} = T(\overline{z}_n^{}) + \sigma \xi_{n+1}^{}$, $\overline{z}_0^{}:=z\in S^1$ of critical circle maps have been studied some time ago by Diaz-Espinoza and de la Llave, who showed for the resulting sum of random variables a central limit theorem and its rate of convergence. Their approach used the renormalization group technique. We will use here Sinai's et al. thermodynamic formalism approach, generalised to circle maps with a break point by Dzhalilov et al., to extend the above results to circle homemorphisms with a break point. This and the sequence of dynamical partitions allows us, following earlier work of Vul at al., to establish a symbolic dynamics for any point~${z\in S^1}$ and to define a~transfer operator whose leading eigenvalue can be used to bound the Lyapunov function. To prove the central limit theorem and its convergence rate we decompose the stochastic sequence via a Taylor expansion in the variables $\xi_i$ into the linear term $L_n^{}(z_0^{})= \xi_n^{}+\sum\limits_{k=1}^{n-1}\xi_k^{}\prod\limits_{j=k}^{n-1} T'(z_j^{})$, ${z_0^{}\in S^1}$ and a higher order term, which is possible in a neighbourhood $A_k^n$ of the points $z_k^{}$, ${k\le n-1}$, not containing the break points of $T^{n}$. For this we construct for a certain sequence~$\{n_m^{}\}$ a series of neighbourhoods $A_k^{n_m^{}}$ of the points $z_k^{}$ which do not contain any break point of the map $T^{q_{n_m^{}}^{}}$, $q_{n_m^{}}^{}$ the first return times of $T$. The proof of our results follows from the proof of the central limit theorem for the linearized process. Keywords: circle map, rotation number, break point, stochastic perturbation, central limit theorem, thermodynamic formalism Citation: Dzhalilov A.,  Mayer D.,  Aliyev A., The Thermodynamic Formalism and the Central Limit Theorem for Stochastic Perturbations of Circle Maps with a Break, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 2, pp. 253-287 DOI:10.20537/nd220208
 Frick P.,  Okatev R.,  Sokoloff D. Abstract The solar 11-year activity cycle is a famous manifestation of magnetic activity of celestial bodies. The physical nature of the solar cycle is believed to be large-scale magnetic field excitation in the form of a wave of a quasi-stationary magnetic field propagating from middle solar latitudes to the solar equator. The power spectrum of solar magnetic activity recorded in sunspot data and underlying solar dynamo action contains quite a stable oscillation known as the 11-year cycle as well as the continuous component and some additional weak peaks. We consider a loworder model for the solar dynamo. We show that in some range of governing parameters this model can reproduce spectra with pronounced dominating frequency and wide spectral peaks in the low-frequency region. The spectra obtained are qualitatively similar to the observed solar activity spectrum. Keywords: solar activity, solar dynamo, low-order models, spectral properties Citation: Frick P.,  Okatev R.,  Sokoloff D., Spectral Properties of Low-order Dynamo Systems, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 2, pp. 289-296 DOI:10.20537/nd220209
 Lee K.,  Rojas A. Abstract In this paper we study the almost shadowable measures for homeomorphisms on compact metric spaces. First, we give examples of measures that are not shadowable. Next, we show that almost shadowable measures are weakly shadowable, namely, that there are Borelians with a measure close to 1 such that every pseudo-orbit through it can be shadowed. Afterwards, the set of weakly shadowable measures is shown to be an $F_{\sigma \delta}$ subset of the space of Borel probability measures. Also, we show that the weakly shadowable measures can be weakly* approximated by shadowable ones. Furthermore, the closure of the set of shadowable points has full measure with respect to any weakly shadowable measure. We show that the notions of shadowableness, almost shadowableness and weak shadowableness coincide for finitely supported measures, or, for every measure when the set of shadowable points is closed. We investigate the stability of weakly shadowable expansive measures for homeomorphisms on compact metric spaces. Keywords: Shadowable measures, Shadowable points, Homeomorphism Citation: Lee K.,  Rojas A., On Almost Shadowable Measures, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 2, pp. 297-307 DOI:10.20537/nd220210
 Ghosh U.,  Das T.,  Sarkar S. Abstract A brief outline of the derivation of the time-fractional nonlinear Schrödinger equation (NLSE) is furnished. The homotopy analysis method (HAM) is applied to study time-fractional NLSE with three separate trapping potential models that we believe have not been investigated yet. The first potential is a double cosine potential $[V(x)=V_1^{}\cos x+V_2^{}\cos 2x]$, the second one is the Morse potential $[V(x)=V_1^{}e^{-2\beta x}+V_2^{}e^{-\beta x}]$, and a hyperbolic potential $[V(x)=V_0^{}\tanh(x)sech(x)]$ is taken as the third model. The fractional derivatives and integrals are described in the Caputo and Riemann Liouville sense, respectively. The solutions are given in the form of convergent series with easily computable components. A physical analysis with graphical representations explicitly reveals that HAM is effective and convenient for solving nonlinear differential equations of fractional order. Keywords: time fractional nonlinear Schrödinger equation (NLSE), homotopy analysis method (HAM), Caputo derivative, Riemann – Liouville fractional integral operator, trapping potential Citation: Ghosh U.,  Das T.,  Sarkar S., Homotopy Analysis Method and Time-fractional NLSE with Double Cosine, Morse, and New Hyperbolic Potential Traps, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 2, pp. 309-328 DOI:10.20537/nd220211