The mechanism of formation of multistability in a van der Pol generator with time-delayed feedback is studied. It is found that the developed multistability is formed through a sequence of two types of bifurcations: supercritical Andronov–Hopf bifurcation and subcritical Neimark–Sacker bifurcation. With variation of the control parameters, the fixed point repeatedly undergoes supercritical Andronov–Hopf bifurcation, which leads to an increase in the number of saddle cycles. The limit cycles acquire stability after a number of subcritical Neimark– Sacker bifurcations. The dynamics of the system is studied in a wide range of control parameters values.

The article deals with the process of fluctuations of a liquid droplet of a small volume lying on a vibrating hydrophobic rigid substrate. The study is carried out by the numerical simulation method of Euler fluid volume (Volume of Fluid — VoF). We study problems of accounting for dynamic changes in the contact angle at the triple point of the liquid-substrate-to-air as well as the impact of changes in the range of the contact angle on the processes that accompany the forced oscillations of the drop. Particular attention is paid to topological features formed in a drop of internal flows. The connection between the interaction of different surface effects, transformation of internal flows, the size limit changes in the contact angle of the substrate and the phase fluctuations are considered in detail. All numerical results are compared with experimental data.

This paper is concerned with the problem of the initial stage of motion of a rigid body in a perturbed liquid when its speed decreases under the linear law. A special feature of this problem is that at large accelerations there are areas of low pressure near the body and attached cavities are formed. Generally the zone of separation is an incoherent set. An important aspect of this study is the problem definition with boundary conditions like inequalities on the basis of which initial zones of separation of particles of the liquid and forms of internal free borders of the liquid on small times are defined. An example is considered in which the initial perturbation of the liquid is caused by a continuous dispersal of the circular cylinder under the free surface of the heavy liquid. A special asymptotic method (like the alternating method of Schwartz) based on the assumption that the free surface of the liquid is at large distances from the floating body is applied to the solution of the last problem.

Long-term aperiodic oscillatory phase transitions of gas – solution and solution – crystal have been reproduced experimentally in the ensemble of aqueous droplets with a dissolved component. The new observations can simplify the dynamic model of phase transitions of the oscillating mode considered before. It reduces the number of independent variables and the need to consider the formation of metastable phases. The model establishes a relation between the speed and direction of solvent flow in the gas and the vapor pressure in the drop neighborhood, temperature of the drop, the rate of change in temperature, the rate of change in volumes of the drop and the crystal. The analysis has revealed that the presence of crystalline phases in the system causes at least two singularities (bifurcations) of the chemical potential of the volatile component with respect to its quantities in the drop.

We investigate the stability of motion of the Maxwell pendulum in a uniform gravity field [1, 2]. The threads on which the axis and the disk of the pendulum have been suspended are assumed to be weightless and inextensible, and the characteristic linear size of the disk is assumed to be small compared to the lengths of threads.
In the unperturbed motion the angle the threads make with the vertical is zero, and the disk moves along the vertical and rotates around its horizontal axis. The nonlinear problem of stability of this motion is solved with respect to small deviations of the threads from the vertical.
By means of canonical transformations and the Poincar´e section surface method, the problem is reduced to the study of stability of the fixed point of the area-preserving mapping of the plane into itself. In the space of dimensionless parameters of the problem, regions of stability and instability are found.

The dynamics of a vibrating ring microgyroscope resonator with open-loop and closed feedback is investigated. We use a mathematical model of forced oscillations for thin elastic resonator, taking into account the nonlinearity coefficient, uneven stiffness, difference in Q-factors and control impact parameters. Using the Krylov–Bogolyubov averaging method, the resonator dynamics in slow variables measured by microgyroscope electronics has been investigated. Formulas with algorithmic compensation of the above defects for determining the angular velocity of the resonator under nonlinear oscillations and without feedback have been obtained. Control signals taking into account the defects are presented for feedback of the microgyroscope operating in the compensation mode of the angular velocity sensor. Numerical modeling of angular velocity determination in the operation modes considered has been carried out.

The problem of the motion of a particle in the gravity field of a homogeneous dumbbell-like body composed of a pair of intersecting balls, whose radii are, in general, different, is studied. Approximation for the Newtonian potential of attraction is obtained. Relative equilibria and their properties are studied under the assumption of uniform rotation of the dumbbells.

We investigate strange nonchaotic self-oscillations in a dissipative system consisting of three mechanical rotators driven by a constant torque applied to one of them. The external driving is nonoscillatory; the incommensurable frequency ratio in vibrational-rotational dynamics arises due to an irrational ratio of diameters of the rotating elements involved. It is shown that, when losing stable equilibrium, the system can demonstrate two- or three-frequency quasi-periodic, chaotic and strange nonchaotic self-oscillations. The conclusions of the work are confirmed by numerical calculations of Lyapunov exponents, fractal dimensions, spectral analysis, and by special methods of detection of a strange nonchaotic attractor (SNA): phase sensitivity and analysis using rational approximation for the frequency ratio. In particular, SNA possesses a zero value of the largest Lyapunov exponent (and negative values of the other exponents), a capacitive dimension close to “2” and a singular continuous power spectrum. In general, the results of this work shed a new light on the occurrence of strange nonchaotic dynamics.

This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario of how one of them arises via a sequence of perioddoubling bifurcations. In addition, we analyze the dynamics of the system in absolute space and show that in the presence of strange attractors in the system the behavior of the point of contact considerably depends on the characteristics of the attractor and can be both chaotic and nearly quasi-periodic.