The present paper is devoted to the study of the dynamics of mappings commuting with pseudo-Anosov surface homeomorphisms. It is proved that the centralizer of a pseudo-Anosov homeomorphism $P$ consists of pairwise nonhomotopic mappings, each of which is a composition of a power of the pseudo-Anosov mapping and a periodic homeomorphism. For periodic mappings commuting with $P$, it is proved that their number is finite and does not exceed the number $N_P^{}$, which is equal to the minimum among the number of all separatrices related to saddles of the same valency of $P$-invariant foliations. For a periodic homeomorphism $J$ lying in the centralizer of $P$, it is also shown that, if the period of a point is equal to half the period of the homeomorphism $J$, then this point is located in the complement of the separatrices of saddle singularities. If the period of the point is less than half the period of $J$, then this point is contained in the set of saddle singularities. In addition, it is proved that there exists a monomorphism from the group of periodic maps commuting with a pseudo-Anosov homeomorphism to the symmetric group of degree $N_P^{}$. Each permutation from the image of the monomorphism is represented as a product of disjoint cycles of the same length. Furthermore, it is deduced that a pseudo-Anosov homeomorphism with the trivial centralizer exists on each orientable closed surface of genus greater than $2$. As an application of the results related to the structure of the centralizer of pseudo-Anosov homeomorphisms to their topological classification, it is proved that there are no pairwise distinct homotopic conjugating mappings for topologically conjugated pseudo-Anosov homeomorphisms.

Spatially distributed integro-differential systems of equations with periodic boundary conditions are considered. In applications, such systems arise as limiting ones for some nonlinear fully coupled ensembles. The simplest critical cases of zero and purely imaginary eigenvalues in the problem of stability of the zero equilibrium state are considered.
In these two situations, quasinormal forms are constructed, for which the question of the existence of piecewise constant solutions is studied. In the case of a simple zero root, the conditions for the stability of these solutions are determined. The existence of piecewise constant solutions with more than one discontinuity point is shown. An algorithm for calculating solutions of the corresponding boundary value problem by numerical methods is presented. A numerical experiment is performed, confirming the analytical constructions.

In this work, by a dynamical system we mean a pair $(S, \,X)$, where $S$ is either a pseudogroup of local diffeomorphisms, or a transformation group, or a smooth foliation of the manifold $X$. The groups of transformations can be both discrete and nondiscrete. We define the concepts of attractor and global attractor of the dynamical system $(S, \,X)$ and investigate the properties of attractors and the problem of the existence of attractors of dynamical systems $(S, \,X)$. Compactness of attractors and ambient manifolds is not assumed. A property of the dynamical system is called transverse if it can be expressed in terms of the orbit space or the leaf space (in the case of foliations). It is shown that the existence of an attractor of a dynamical system is a transverse property. This property is applied by us in proving two subsequent criteria for the existence of an attractor (and global attractor): for foliations of codimension $q$ on an $n$-dimensional manifold, $0 < q < n$, and for foliations covered by fibrations. A criterion for the existence of an attractor that is a minimal set for an arbitrary dynamical system is also proven. Various examples of both regular attractors and attractors of transformation groups that are fractals are constructed.

We consider nonconservative quasi-periodic perturbations of two-dimensional Hamiltonian systems with nonmonotonic rotation. The peculiarity of these systems is that degenerate resonances can take place. Special focus is on those systems for which the corresponding autonomous perturbed system has a structurally stable limit cycle. If the cycle appears in the neighborhood of a nonresonance phase curve, then it corresponds to an invariant torus in the initial system. In this regard, the problem of synchronization arises when the invariant torus passes through a resonance zone. In this paper, we distinguish a class of perturbations (the so-called parametric perturbations) under which synchronization can be violated. Also, we introduce the concept of generalized synchronization and give conditions for this type of synchronization to occur. As an example, we study a Duffing-like equation with an asymmetric potential function.

This paper deals with mechanical systems subject to nonlinear nonholonomic constraints. The aspect that is mainly investigated is the formulation of an energy equation, which is deduced from the equations of motion, via the generalization of standard procedures pertaining to simpler cases (holonomic systems). We consider the equations of motion in two different forms (both present in the literature): the first method introduces the Lagrange multipliers, the second one is based on the selection of a certain number of independent velocities. The second procedure generalizes Voronec’s equations for linear kinematic constraints to the case of nonlinear kinematic constraints. The two kinds of equations of motion gives rise to two different ways of writing an energy-type equation, since in the second case only the independent velocities are used and restricted functions corresponding to the Lagrangian and to the Legendre transform consequently appear in the energy equation. We see that the discrepancy between energy and its reduced version disappears if and only if the explicit expressions of the constraints are homogeneous functions of degree 1 with respect to the independent velocities. This property has an effect to various aspects of the problem and produces certain benefits which place nonlinear nonholonomic system near to the linear ones.
From the mathematical point of view, we prove that the homogeneity of the explicit functions is equivalent to the homogeneity of the given constraints with respect to the full set of velocities. This conclusion is in step with previous results, where the conservation of energy in nonlinear nonholonomic system is strictly connected with the homogeneity of the constraint functions. According to the present work, the known result is put into perspective together with other aspects of nonlinear nonholonomic systems, mainly investigating the unifying role exerted by the homogeneity property of the explicit constraint functions. A selection of cases and examples is discussed.

The impact of migration effects on the formation of population distributions is studied. We consider the model of interplay between two species (resident and invader), and apply the theory of cosymmetry to classify different population scenarios. The system of reaction – diffusion – advection equations is used to describe the nonlinear diffusion and taxis because of nonuniform distribution of the resource. The logistic law of growth is taken to model local interaction between species. We consider a one-dimensional habitat with no-flux boundary conditions. Finite-difference discretization with a staggered grid is used for the spatial coordinate and Runge – Kutta integrator to solve the resulting system of ordinary differential equations of large order. A computer experiment is applied to analyze the dynamics of populations and migration fluxes. We numerically build the maps of migration parameters for description scenarios of invasion and competition. It is found that nonlinear diffusion has an influence on invasion because intraspecific taxis compensates nonoptimal migration to resource. Negative coefficients of intraspecific taxis stimulate diffusion for both species and prevent excessive concentration of populations. This aids the coexistence of species as stationary distributions. Different coefficient signs imply the implementation of corresponding stable semipositive solutions. Direct numerical experiments show that the coexistence of species occurs at large positive coefficients of intraspecific taxis. The dependence of the scenario on the initial distributions is established.

This article discusses the mechanism of parallel structure, which includes hinged parallelograms. These mechanisms have a certain peculiarity when composing kinematics equations, consisting in the fact that some of the equations have a linear form. This simplifies the system of coupling equations as a whole. By solving direct and inverse kinematics, we will determine the size and shape of the working area. A method was chosen by solving the inverse kinematics to determine the workspace. The size and shape of the working area of the mechanism under consideration with three degrees of freedom are experimentally determined under given initial conditions. The presence of a large working area allows us to recommend this mechanism for use in various branches of robotics, medicine, simulators, etc. The Jacobian matrix of the coupling equations of the mechanism is written out to determine the singularities.

This paper investigates the trajectories of light beams in a Kerr metric, which describes the gravitational field in the neighborhood of a rotating black hole. After reduction by cyclic coordinates, this problem reduces to analysis of a Hamiltonian system with two degrees of freedom. A bifurcation diagram is constructed and a classification is made of the types of trajectories of the system according to the values of first integrals. Relations describing the boundary of the shadow of the black hole are obtained for a stationary observer who rotates with an arbitrary angular velocity about the axis of rotation of the black hole.

This paper considers a nonlinear fourth-order ordinary differential equation. The study of this class of equations is conducted using an analytical approximation method based on dividing the solution domain into two parts: the region of analyticity and the vicinity of a movable singular point. This work focuses on investigating the equation in the region of analyticity and solving two problems. The first problem is a classical problem in the theory of differential equations: proving the theorem of existence and uniqueness of a solution in the region of analyticity. The structure of the solution in this region takes the form of a power series. To transition from formal series to series converging in a neighborhood of the initial conditions, a modification of the majorant method is used, which is applied in the Cauchy – Kovalevskaya theorem. This method allows determining the domain of validity of the theorem. Within this domain, error estimates for the analytical approximate solution are obtained, enabling the solution to be found with any predefined accuracy. When leaving the domain of the theorem’s validity, analytical continuation is required. To do this, it is necessary to solve the second task of the study: to study the effect of perturbation of the initial data on the structure of the analytical approximate solution.

This paper is concerned with the motion of an elliptic foil in the field of a fixed point singularity. A complex potential of the fluid flow is constructed, and the forces and the torque which act on the foil from the fluid are obtained. It is shown that the equations of motion of the elliptic foil in the field of a fixed point vortex source can be represented as Lagrange – Euler equations. It is also shown that the system has an additional first integral due to the conservation of the angular momentum. An effective potential of the system under consideration is constructed. For the cases where the singularity is a vortex or a source, unstable relative equilibrium points corresponding to the circular motion of the foil around the singularity are found.

This paper develops the holomorphic regularization method of the Cauchy problem for a special type of Tikhonov system that arises in the mathematical modeling of wave solidstate gyroscope dynamics. The Tikhonov system is a system of differential equations a part of which is singularly perturbed. Unlike other asymptotic methods giving approximations in the form of asymptotically converging series, the holomorphic regularization method allows one to obtain solutions of nonlinear singularly perturbed problems in the form of series in powers of a small parameter converging in the usual sense. Also, as a result of applying the holomorphic regularization method, merged formulas for an approximate solution are deduced both in the boundary layer and outside it. These formulas allow a qualitative analysis of the approximate solution on the entire time interval including the boundary layer.
This paper consists of two sections. In Section 1, the holomorphic regularization method of the Cauchy problem for a special type of Tikhonov system is developed. The special type of Tikhonov system means the following: singularly perturbed equations are linear in the variables included in them with derivatives, the matrix of the singularly perturbed part of the system is diagonal, the remaining equations have separate linear and nonlinear parts. An algorithm for deriving an approximate solution to the Cauchy problem for the Tikhonov system of special type by using the holomorphic regularization method is presented. In Section 2, the mathematical model describing in interconnected form the mechanical oscillations of the gyroscope resonator and the electrical processes in the oscillation control circuit is considered. The algorithm for deriving an approximate solution proposed in Section 1 is used. Formulas for an approximate solution taking into account the structure of the Tikhonov system are deduced.

An analysis is made of the motion of a particle on a rotating disk equipped with rectilinear blades for different types of particle collisions with a blade. The dynamics of particle motion is considered using equations describing the dynamics of the relative motion of a particle and is studied for the cases of inelastic and elastic interaction with the blade when the coefficient of friction of the particle against the disk has a value that is the most probable from a practical point of view. The impact interaction of the particle with the blade is studied using the Routh hypothesis, which relates the normal and tangential components of the impact impulse by a dependence similar to the Amonton – Coulomb dependence between the normal and tangential components of the interaction force of two rough bodies. The dynamical coefficient of impact friction is taken to be equal to half the kinematic coefficient of friction of the particle against the disk. After an impact, depending on the values of the above-mentioned coefficients and the angle of inclination of the blade, the particle may or may not possess a velocity component tangential and (or) normal to the surface of the blade. In addition, depending on the overall dimensions of the disk and the inclination angle of the blade the moving particle can repeatedly enter into impact interaction with the blade. In this paper we consider all four patterns of motion of the particle after its first and subsequent impacts against the blade of the rotating disk, present graphs of time dependences of the projections of the velocity of the relative motion of the particle, and plot trajectories of the relative motion of the particle for two practically important cases of installation of a blade on the disk’s surface which ensure the initial collision angles equal to 0${}^\circ$ and 30${}^\circ$. The proposed model is also used to investigate the changes in the antitorque moment acting on the disk. For this purpose, in each of the cases we calculate the resulting moment of impact impulse and the impulse of friction torque throughout the motion of the particle on the disk’s surface. We present the results of numerical experiments and use them to give recommendations on the optimization of mechanical systems of this type.

Knots naturally appear in continuous dynamical systems as flow periodic trajectories. However, discrete dynamical systems are also closely connected with the theory of knots and links. For example, for Pixton diffeomorphisms, the equivalence class of the Hopf knot, which is the orbit space of the unstable saddle separatrix in the manifold $\mathbb{S}^2\times \mathbb{S}^1$, is a complete invariant of the topological conjugacy of the system. In this paper we distinguish a class of three-dimensional Morse – Smale diffeomorphisms for which the complete invariant of topological conjugacy is the equivalence class of a link in $\mathbb{S}^2\times \mathbb{S}^1$.
We prove that, if $M$ is a link complement in $\mathbb{S}^3$, or a handlebody $H_g^{}$ of genus $g\geqslant 0$, or a closed, connected, orientable 3-manifold, then the set of equivalence classes of tame links in $M$ is countable. As a corollary, we prove that there exists a countable number of equivalence classes of tame links in $\mathbb{S}^2\times \mathbb{S}^1$. It is proved that any essential link can be realized by a diffeomorphism of the class under consideration.