The intermittent behavior near the boundary of the noise-induced synchronization regime is studied. «On-off» intermittency is shown to take place in this case. The observed phenomenon is illustrated by considering both model systems with discrete time and flow dynamical systems being under influence of the common source of noise.

Nonlocal generalizations of nonlinear wave equation arise in numerous physical applications. It is known that switching from local to nonlocal description may result in new features of the problem and new types of solutions. In this paper the author analyses the dimension of the set of travelling wave solutions for а nonlocal nonlinear wave equation. The nonlocality is represented by the convolution operator which replaces the second derivative in the dispersion term. The results have been obtained for the case where the nonlinearity is bounded, and the kernel of the convolution operator is represented by a sum of exponents with weights (so-called E-type kernel). In the simplest particular case, (so-called Kac—Baker kernel) it is shown that the solutions of this equation form a 3-parametric set (assuming the equivalence of the solutions which differ by a shift with respect to the independent variable). Then it is shown that in the case of the general E-type kernel the 3-parametric set of solutions also exists, generically, under some additional restrictions. The word «generically» in this case means some transversality condition for intersection of some manifolds in a properly defined phase space.

In this paper class $MS(M^3)$ of Morse–Smale diffeomorphisms (cascades) given on connected closed orientable 3-manifolds are considered. For a diffeomorphism $f \in MS(M^3)$ it is introduced a notion scheme $S_f$, which contains an information on the periodic data of the cascade and a topology of embedding of the sepsrstrices of the saddle points. It is established that necessary and sufficient condition for topological conjugacy of diffeomorphisms $f$, $f’ \in MS(M^3)$ is the equivalence of the schemes $S_f$, $S_f’$.

The behavior of the two-gap elliptic solutions of the Boussinesq and the KdV equations was examined. These solutions were constructed by the $n$-sheet covering over a torus $(n \leqslant 3)$. It was shown that the shape of the two-gap elliptic solutions depends on $n$ and doesn’t depend on the kind of the nonlinear wave equation.

Within the context of the weakly nonlinear approach, the leading nonlinear contribution to the development of unstable disturbances in shear flows should be made by resonant three-wave interaction, i.e., the interaction of triplets of such waves that have a common critical layer (CL), and their wave vectors form a triangle. Surprisingly, the subharmonic resonance proves to be the only such interaction that has been studied so far. The reason for this is that in many cases, the requirement of having a common CL produces too rigid selection of waves which can participate in the interaction. We show that in a broad spectral range, Holmboe waves in sharply stratified shear flows can have a common CL, and examine the evolution of small ensembles consisting of several interrelated triads of those waves. To do this, the evolution equations are derived which describe the three-wave interaction and have the form of nonlinear integral equations. Analytical and numerical methods are both used to find their solutions in different cases, and it is shown that at the nonlinear stage disturbances increase, as a rule, explosively.

The integrable and nonintegrable motion of a vortex pair, which consists of two vortices of arbitrary intensities, embedded inside a steady and periodic external deformation flow is studied. In the general case, such an external deformation flow impacts asymmetrically on the vortex pair, which results in nonconservation of motion invariants: the linear momentum and the angular momentum. An analytical expression for the linear momentum, which gives an opportunity to reduce the initial system with 2.5 degrees of freedom to a system with 1.5 degrees of freedom, is obtained. For the steady state of a constant deformation flow the integrability of the dipole motion is shown for any initial vortices positions and intensities of vortices, and for arbitrary values of shear and rotation of the deformation flow.

We study a particle motion along a cable with ends fixed in a precessed rigid body. Such cable called «the leier» is a model of space elevator for a dynamically symmetric asteroid. (The Dutch term «leier» means the rope with fixed ends). In this paper we find two integrable cases of the particle motion equations (for zero and right nutation angle) Phase portraits for integrable situations are built taking into account conditions of motion with the tense cable and assuming the body gravitation is close to gravitational field of two equal point masses that are in the axis of dynamical symmetry. Using «the Generalized Restricted Circular Problem of Three Bodies» by V. V. Beletsky, we study the particle equilibria on the leier in the plane containing the body mass center and being perpendicular to the precession axis for all possible nutation angles. Some facts on these equilibria stability are formulated.

We consider the problem of explicit integration and bifurcation analysis for two systems of nonholonomic mechanics. The first one is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetrical ball on a horizontal plane. The second problem is on the motion of rigid body in a spherical support. We explicitly integrate this problem by generalizing the transformation which Chaplygin applied to the integration of the problem of the rolling ball at a non-zero constant of areas. We consider the geometric interpretation of this transformation from the viewpoint of a trajectory isomorphism between two systems at different levels of the energy integral. Generalization of this transformation for the case of dynamics in a spherical support allows us to integrate the equations of motion explicitly in quadratures and, in addition, to indicate periodic solutions and analyze their stability. We also show that adding a gyrostat does not lead to the loss of integrability.

We review previous investigations concerning the terminal motion of disks sliding and spinning with uniform dry friction across a horizontal plane. Previous analyses show that a thin circular ring or uniform circular disk of radius $R$ always stops sliding and spinning at the same instant. Moreover, under arbitrary nonzero initial values of translational speed $v$ and angular rotation rate $ω$, the terminal value of the speed ratio $ε_0 = v/Rω$ is always 1.0 for the ring and 0.653 for the uniform disk. In the current study we show that an annular disk of radius ratio $η = R_2/R_1$ stops sliding and spinning at the same time, but with a terminal speed ratio dependent on $η$. For a twotier disk with lower tier of thickness $H_1$ and radius $R_1$ and upper tier of thickness $H_2$ and radius $R_2$, the motion depends on both $η$ and the thickness ratio $λ = H_1/H_2$. While translation and rotation stop simultaneously, their terminal ratio $ε_0$ either vanishes when $k > \sqrt{2/3}$, is a nonzero constant when $1/2 < k < \sqtr{2/3}$, or diverges when $k < 1/2$, where $k$ is the normalized radius of gyration. These three regimes are in agreement with those found by Goyal et al. [S.Goyal, A.Ruina, J.Papadopoulos, Wear 143 (1991) 331] for generic axisymmetric bodies with varying radii of gyration using geometric methods. New experiments with PVC disks sliding on a nylon fabric stretched over a plexiglass plate only partially corroborate the three different types of terminal motions, suggesting more complexity in the description of friction.

Large objects which propel themselves in air or water make use of inertia in the surrounding fluid. The propulsive organ pushes the fluid backwards, while the resistance of the body gives the fluid a forward momentum. The forward and backward momenta exactly balance, but the propulsive organ and the resistance can be thought about as acting separately. This conception cannot be transferred to problems of propulsion in microscopic bodies for which the stresses due to viscosity may be many thousands of times as great as those due to inertia. No case of self-propulsion in a viscous fluid due to purely viscous forces seems to have been discussed.
The motion of a fluid near a sheet down which waves of lateral displacement are propagated is described. It is found that the sheet moves forwards at a rate $2π^2 b^2/λ^2$ times the velocity of propagation of the waves. Here $b$ is the amplitude and $λ$ the wave-length. This analysis seems to explain how a propulsive tail can move a body through a viscous fluid without relying on reaction due to inertia. The energy dissipation and stress in the tail are also calculated.
The work is extended to explore the reaction between the tails of two neighbouring small organisms with propulsive tails. It is found that if the waves down neighbouring tails are in phase very much less energy is dissipated in the fluid between them than when the waves are in opposite phase. It is also found that when the phase of the wave in one tail lags behind that in the other there is a strong reaction, due to the viscous stress in the fluid between them, which tends to force the two wave trains into phase. It is in fact observed that the tails of spermatozoa wave in unison when they are close to one another and pointing the same way.

This paper has been written for a collection of V.A. Steklov’s selected works, which is being prepared for publication and is entitled «Works on Mechanics 1902–1909: Translations from French». The collection is based on V.A. Steklov’s papers on mechanics published in French journals from 1902 to 1909.