 COEDITORSINCHIEF
 Honorary Editor
 Editorial board

Vol. 16, no. 3
Osipov A. S.
Abstract
In this paper, some links between inverse problem methods for the secondorder 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 semiinfinite Jacobi
matrices can be applied to obtain a hierarchy of Volterra lattices, and this approach is compared
with the one based on Magri’s biHamiltonian 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 abovementioned moments.

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
roundoff errors and perturbations not taken into account in the model.

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.

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.

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 biHamiltonian 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.

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.

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 singlefrequency 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 mainorder 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 stepbystep 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.

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 flipover
phenomenon which arises in the experiment with Dzhanibekov’s nut.
