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The population and number dynamics of small mammals in the vicinity of the Torey lakes (Southeast Transbaikalia, Russia) were characterized between 2008 and 2017. The monitoring was carried out against the backdrop of the complete drying out of these largest lakes in the region. The stability of the fauna of small mammals was revealed over an 80-year period, although the structure of communities has changed significantly as a result of changes in moistening of the territory. Xerophilous species of mammals had an advantage in the period of studies in the dry climate phase. Correlation analysis showed a possible link between the population dynamics of some common species of small mammals with the precipitation amount of the current and previous years or with the precipitation amount of some spring (in one case, winter) months.

The fragmentation of metals during megaplastic (severe) deformation is described in terms of a two-defect model. The process kinetics has been studied qualitatively. A general expression for the Lyapunov indices, which determine the stability of limiting (stationary) structures, is derived. Critical conditions for controlling parameters are found, and the diagrams that determine the stability of stationary states are constructed.

Under consideration is the system of partial differential equations describing the dynamics of a two-phase medium. Exact partially invariant solutions of rank 1 and defect 1 of this system are obtained with respect to some four-dimensional subalgebras. The phenomenon of collapse (an instantaneous source) in a two-phase medium is described.

The problem of interval estimation of the time delay of the coupling between oscillatory systems from observed time series is considered. It is shown that the known asymptotic estimates based on the empirical model in the form of a system of first-order phase oscillators and the maximum likelihood formalism can lead to false inferences of the value of the time delay in two typical situations: 1) nonlinear low-dimensional systems whose phases are well-determined but, as a result of significant amplitude fluctuations, the phase approximation is insufficient for describing the dynamics and 2) systems whose phases are defined not quite well because of too large amplitude fluctuations. A method for empirical diagnostics of problematic situations and its modification (coarse estimation) providing a low probability of false inferences in these situation are proposed. The efficiency of the diagnostic criterion and coarse estimation suggested is demonstrated on reference systems with different dynamic properties (linear stochastic oscillators, van der Pol oscillators, and Ressler and Lorenz chaotic systems).

The phase dynamics of a SQUID consisting of Josephson junctions with topologically nontrivial barriers has been studied. Its comparative analysis with the dynamics of a conventional SQUID has been performed. The current–voltage characteristics have been calculated. The dependence of the return current on the magnetic field has been found. It has been shown that the branch of the current–voltage characteristic corresponding to the resonance frequency in the case of the SQUID with nontrivial barriers is displaced by \(\sqrt 2 \) over voltage. This effect can be used for the detection of Majorana fermions.

The phase dynamics of two parallel connected stacks of intrinsic Josephson junctions (JJs) in high temperature superconductors is numerically investigated. The calculations are based on the system of nonlinear differential equations obtained within the CCJJ + DC model, which allows one to determine the general current-voltage characteristic of the system, as well as each individual stack. The processes with increasing and decreasing base currents are studied. The features in the behavior of the current in each stack of the system due to the switching between the states with rotating and oscillating phases are analyzed.

The need to estimate the delay times between elements of coupled oscillatory ensembles is frequently encountered in the analysis of systems of various natures, for which only separate records (time series) of the observed oscillations are available. The possibility of obtaining these estimates in the case of relatively short time series recorded in the presence of noise is considered. Using the well-known point estimator based on a model of the phase dynamics, an interval estimator is proposed, a correction that eliminates a bias of the point estimate is introduced, and a formula for the confidence interval is derived based on the maximum likelihood formalism. Possibilities of the proposed approach are illustrated by the results of numerical experiments in systems of coupled oscillators.

The detailed investigation of the phase dynamics and the I–V curves in the system of coupled Josephson junctions have been carried out. The superconducting, quasiparticle, diffusion, and displacement currents have been calculated as functions of the total current through the system. The role of the diffusion current in the formation of the I–V curves has been studied and the influence of this quantity on the I–V curve branching and the magnitude of the return current has been revealed. The calculation results agree qualitatively with the experimental data.

The relations between the processes occurring in the equatorial latitudes of the Pacific and Atlantic oceans were studied on the basis of the Granger causality analysis and a simulation of phase dynamics using the indices of the El Nino-Southern Oscillation (ENSO) and the equatorial Atlantic mode (EAM). Data on the monthly means of the sea-surface temperature over the period 1870–2006 for the Nino 3 (5° S-5° N, 150° W-90° W) and Nino 3.4 (5° S-5° N, 170° W-120° W) regions in the Pacific and the Atlantic 3 region (20° W-0, 3° S-3° N) in the Atlantic Ocean were used as the ENSO and EAM indices. The statistically significant influence of the EAM on the ENSO is noted. The lag time of this influence is estimated at two months. No significant reverse effect is revealed. An increase in the EAM’s influence on the ENSO was observed in the second half of the 20th century.

The melting of an ultrathin lubricant film at friction between atomically smooth surfaces is studied with allowance for fluctuations of its temperature, which are described by the Ornstein-Uhlenbeck process. The behavior of the most probable types of shear stresses arising in the lubricant is considered, and phase diagrams for second- and first-order phase transformations (the melting of an amorphous lubricant and that of a crystal- line lubricant, respectively) are constructed. It is shown that, in the former case, lubricant temperature fluctua- tions lead to the formation of a stick-slip friction domain separating the domains of dry and sliding friction, which is typical of first-order transitions. In the latter case, three domains of stick-slip friction arise, which mark the transitions between dry friction and metastable and stable sliding friction. As the time of correlation of lubricant temperature fluctuations gets longer, the temperature of rubbing surfaces rises to the point where sliding friction sets in.

One of the most sensitive methods for determining the presence and character of coupling between two oscillators is based on the modeling of their phase dynamics. This method is applicable if the observed oscillations are not synchronous and the length of available time series is no less than 50 characteristic periods. Now it is established that the method can be applied to still shorter time series with a length of 20 to 50 periods, provided that the estimated phase coherence coefficient (measuring the degree of synchronization) does not exceed a certain threshold dependent on the time series length and the frequency detuning. This result expands the possibilities of detecting the coupling between oscillatory systems under conditions of nonstationary signals and deficient data.

A new method is proposed for the investigation of transitions from synchronous dynamics of coupled oscillators to asynchronous regimes attained upon synchronization breakage by a sequence of short phase resetting pulses. The possibility to use the proposed method for determining the coefficient of coupling between interacting systems from an analysis of the signals generated by these systems is considered.

The statistics of the instantaneous phase of oscillations in dynamic systems with a noisy limit cycle is compared to the statistics of the instantaneous phase of oscillations in dynamic systems with a spiral chaotic attractor. Simulation of the phase dynamics of chaotic self-sustained oscillations by a Wiener process is considered. The results provided by various methods of determination of the instantaneous phase are analyzed.

New types of stable discrete solitons are discovered. They represent the first example of asymmetric dark solitons and shock waves with a nonzero background. Both types of solutions exhibit a strong intrinsic phase dynamics. Their domains of existence and criteria of stability are identified. Numerical experiments support the analytical findings.