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2013
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Vol. 16, no. 3

Osipov A. S.
Abstract
In this paper, some links between inverse problem methods for the second-order difference operators and nonlinear dynamical systems are studied. In particular, the systems of Volterra type are considered. It is shown that the classical inverse problem method for semi-infinite Jacobi matrices can be applied to obtain a hierarchy of Volterra lattices, and this approach is compared with the one based on Magri’s bi-Hamiltonian formalism. Then, using the inverse problem method for nonsymmetric difference operators (which amounts to reconstruction of the operator from the moments of itsWeyl function), the hierarchies of Volterra and Toda lattices are studied. It is found that the equations of Volterra hierarchy can be transformed into their Toda counterparts, and this transformation can be easily described in terms of the above-mentioned moments.
Keywords: inverse spectral problems, difference operators, Jacobi matrices, Volterra lattices, Toda lattices
Citation: Osipov A. S., Inverse Spectral Problems for Second-Order Difference Operators and Their Application to the Study of Volterra Type Systems, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 3, pp. 397-419
DOI:10.20537/nd200301
Kiselev O. M.
Abstract
The dynamics of an inverted wheeled pendulum controlled by a proportional plus integral plus derivative action controller in various cases is investigated. The properties of trajectories are studied for a pendulum stabilized on a horizontal line, an inclined straight line and on a soft horizontal line. Oscillation regions on phase portraits of dynamical systems are shown. In particular, an analysis is made of the stabilization of the pendulum on a soft surface, modeled by a differential inclusion. It is shown that there exist trajectories tending to a semistable equilibrium position in the adopted mathematical model. However, in numerical simulations, as well as in the case of real robotic devices, such trajectories turn into a limit cycle due to round-off errors and perturbations not taken into account in the model.
Keywords: pendulum, control, stability, differential inclusion
Citation: Kiselev O. M., Control of an Inverted Wheeled Pendulum on a Soft Surface, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 3, pp. 421-436
DOI:10.20537/nd200302
Kubyshkin E. P.,  Moriakova A. R.
Abstract
This paper deals with bifurcations from the equilibrium states of periodic solutions of the Ikeda equation, which is well known in nonlinear optics as an equation with a delayed argument, in two special cases that have not been considered previously. Written in a characteristic time scale, the equation contains a small parameter with a derivative, which makes it singular. Both cases share a single mechanism of the loss of stability of equilibrium states under changes of the parameters of the equation associated with the passage of a countable number of roots of the characteristic equation through the imaginary axis of the complex plane, which are in this case in certain resonant relations. It is shown that the behavior of solutions of the equation with initial conditions from fixed neighborhoods of the studied equilibrium states in the phase space of the equation is described by countable systems of nonlinear ordinary differential equations that have a minimal structure and are called the normal form of the equation in the vicinity of the studied equilibrium state. An algorithm for constructing such systems of equations is developed. These systems of equations allow us to single out one “fast” variable and a countable number of “slow” variables, which makes it possible to apply the averaging method to the systems of equations obtained. Equilibrium states of the averaged system of equations of “slow” variables in the original equation correspond to periodic solutions of the same nature of sustainability. In the special cases under consideration, the possibility of simultaneous bifurcation from equilibrium states of a large number of stable periodic solutions (multistability bifurcation) and evolution of these periodic solutions to chaotic attractors with changing bifurcation parameters is shown. One of the special cases is associated with the formation of paired equilibrium states (a stable and an unstable one). An analysis of bifurcations in this case provides an explanation of the formation of the “boiling points of trajectories”, when a periodic solution arises “out of nothing” at some point in the phase space under changes of the parameters of the equation and quickly becomes chaotic.
Keywords: Ikeda equation, periodic solutions, bifurcation of multistability, chaotic multistability
Citation: Kubyshkin E. P.,  Moriakova A. R., Analysis of Special Cases in the Study of Bifurcations of Periodic Solutions of the Ikeda Equation, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 3, pp. 437-451
DOI:10.20537/nd200303
Borisov A. V.,  Mikishanina E. A.
Abstract
This work is devoted to the study of the dynamics of the Chaplygin ball with variable moments of inertia, which occur due to the motion of pairs of internal material points, and internal rotors. The components of the inertia tensor and the gyrostatic momentum are periodic functions. In general, the problem is nonintegrable. In a special case, the relationship of the problem under consideration with the Liouville problem with changing parameters is shown. The case of the Chaplygin ball moving from rest is considered separately. Poincaré maps are constructed, strange attractors are found, and the stages of the origin of strange attractors are shown. Also, the trajectories of contact points are constructed to confirm the chaotic dynamics of the ball. A chart of dynamical regimes is constructed in a separate case for analyzing the nature of strange attractors.
Keywords: Chaplygin ball, Poincaré map, strange attractor, chart of dynamical regimes
Citation: Borisov A. V.,  Mikishanina E. A., Dynamics of the Chaplygin Ball with Variable Parameters, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 3, pp. 453-462
DOI:10.20537/nd200304
Bekir A.,  Shehata M.,  Zahran E.
Abstract
In this article, we employ the Painlevé approach to realize the solitary wave solution to three distinct important equations for the shallow water derived from the generalized Camassa – Holm equation with periodic boundary conditions. The first one is the Camassa – Holm equation, which is the main source for the shallow water waves without hydrostatic pressure that describes the unidirectional propagation of waves at the free surface of shallow water under the influence of gravity. While the second, the Novikov equation as a new integrable equation, possesses a bi-Hamiltonian structure and an infinite sequence of conserved quantities. Finally, the third equation is the (3 + 1)-dimensional Kadomtsev – Petviashvili (KP) equation. All the ansatz methods with their modifications, whether they satisfy the balance rule or not, fail to construct the exact and solitary solutions to the first two models. Furthermore, the numerical solutions to these three equations have been constructed using the variational iteration method.
Keywords: Camassa – Holm equation, Novikov – Veselov equation, (3 + 1)-dimensional Kadomtsev – Petviashvili (KP) equation, Painlevé approach, traveling wave solutions, numerical solutions
Citation: Bekir A.,  Shehata M.,  Zahran E., Comparison Between the Exact Solutions of Three Distinct Shallow Water Equations Using the Painlevé Approach and Its Numerical Solutions, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 3, pp. 463-477
DOI:10.20537/nd200305
Morozov A. Y.,  Reviznikov D. L.
Abstract
In solving applied and research problems, there often arise situations where certain parameters are not exactly known, but there is information about their ranges. For such problems, it is necessary to obtain an interval estimate of the solution based on interval values of parameters. In practice, the dynamic systems where bifurcations and chaos occur are of interest. But the existing interval methods are not always able to cope with such problems. The main idea of the adaptive interpolation algorithm is to build an adaptive hierarchical grid based on a kdtree where each cell of adaptive hierarchical grid contains an interpolation grid. The adaptive grid should be built above the set formed by interval initial conditions and interval parameters. An adaptive rebuilding of the partition is performed for each time instant, depending on the solution. The result of the algorithm at each step is a piecewise polynomial function that interpolates the dependence of the problem solution on the parameter values with a given precision. Constant grid compaction will occur at the corresponding points if there are unstable states or dynamic chaos in the system; therefore, the minimum cell size is set. The appearance of such cells during the operation of the algorithm is a sign of the presence of unstable states or chaos in a dynamic system. The effectiveness of the proposed approach is demonstrated in representative examples.
Keywords: interval ODE, bifurcations, instability, dynamic chaos, adaptive interpolation algorithm, kd-tree
Citation: Morozov A. Y.,  Reviznikov D. L., Modeling of Dynamic Systems with Interval Parameters in the Presence of Singularities, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 3, pp. 479-490
DOI:10.20537/nd200306
Smirnov V. V.,  Manevitch L. I.
Abstract
We present the complex envelope variable approximation (CEVA) as a useful and compact method for analysis of essentially nonlinear dynamical systems. The basic idea is that the introduction of complex variables, which are analogues of the creation and annihilation operators in quantum mechanics, considerably simplifies the analysis of a number of nonlinear dynamical systems. The first stage of the procedure, in fact, does not require any additional assumptions, except for the proposition of the existence of a single-frequency stationary solution. This allows us to study both the stationary and nonstationary dynamics even in the cases when there are no small parameters in the initial problem. In particular, the CEVA method provides an analysis of nonlinear normal modes and their resonant interactions in discrete systems for a wide range of oscillation amplitudes. The dispersion relations depending on the oscillation amplitudes can be obtained in analytical form for both the conservative and the dissipative nonlinear lattices in the framework of the main-order approximation. In order to analyze the nonstationary dynamical processes, we suggest a new notion — the “slow” Hamiltonian, which allows us to generate the nonstationary equations in the slow time scale. The limiting phase trajectory, the bifurcations of which determine such processes as the energy localization in the nonlinear chains or the escape from the potential well under the action of external forces, can be also analyzed in the CEVA. A number of complex problems were studied earlier in the framework of various modifications of the method, but the accurate formulation of the CEVA with the step-by-step illustration is described here for the first time. In this paper we formulate the CEVA’s formalism and give some nontrivial examples of its application.
Keywords: nonlinear dynamical systems, asymptotic methods, nonlinear normal modes, limiting phase trajectory, complex envelope approximation
Citation: Smirnov V. V.,  Manevitch L. I., Complex Envelope Variable Approximation in Nonlinear Dynamics, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 3, pp. 491-515
DOI:10.20537/nd200307
Zhuravlev V. F.,  Rozenblat G. M.
Abstract
This paper presents secure upper and lower estimates for solutions to the equations of rigid body motion in the Euler case (in the absence of external torques). These estimates are expressed by simple formulae in terms of elementary functions and are used for solutions that are obtained in a neighborhood of the unstable steady rotation of the body about its middle axis of inertia. The estimates obtained are applied for a rigorous explanation of the flip-over phenomenon which arises in the experiment with Dzhanibekov’s nut.
Keywords: Euler top, permanent (steady) rotation, middle axis of inertia, estimates of solutions to differential equations
Citation: Zhuravlev V. F.,  Rozenblat G. M., Estimates of Solutions During Motion of the Euler –Poinsot Top and Explanation of the Experiment with Dzhanibekov’s Nut, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 3, pp. 517-525
DOI:10.20537/nd200308
Abstract
Citation: In memory of Professor Leonid I. Manevitch, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 3, pp. 527-528
DOI:10.20537/nd200309