Applied Problems of Nonlinear Oscillation and Wave Theory

The paper deals the external resonator effect on collective electron dynamics in the semiconductor superlattice. Numerical simulations have shown that semiconductor superlattice which is characterized by only the regular oscillations in autonomous regime demonstrates chaotic microwave oscillations in the case of interaction with the external electrodynamical system. This phenomena gives a strong potential for novel applications of semiconductor superlattice requiring microwave chaotic signals.

Experimental investigations results of the ring self-oscillatory system based on a ferromagnetic film at three-wave interactions were considered. The model describing this system was constructed. The typical regimes of a generation, including generation of the chaotic microwave train were calculated with the help of the constructed model. The numerical simulations and experimental results had a good agreement.

Experimental and theoretical study of nonlinear dynamics and acoustic signals generated by periodic impacts of corundum probe on the solid surface are conducted. In the work two models are considered for the description of experiments: the analytical model based on the laws of conservation of energy and momentum; the model based on the numerical solution of the nonlinear equation of probe motion. It is shown that the acoustic signal amplitude increases in direct proportion to the oscillations probe amplitude.

The influence of slow processes (random or regular) on the probability distribution of fast random processes is considered. We show that such influence is universal for all random processes, and in some cases this universality is of the multifractal character. As an example we consider stochastic resonance.

In the Pliocene (about five – two million years before present) global climate fluctuated with a period corresponding well 41-thousand-year cycle of changes in the Earth’s axis inclination to the ecliptic plane. Then, this period has disappeared, despite the fact that the 41-thousand-year cycle even slightly increased its scope and, therefore, the response to it would have only strengthened. By analyzing paleoclimatic series covering the Pliocene and subsequent Pleistocene, we show that the response of the climate system simply became unstable and therefore unobservable.

Currently, the method of nonlinear Granger causality is actively used in many applications in medicine, biology, physics, to identify the coupling between objects from the records of their oscillations (time series) using forecasting models. In this paper the impact of choosing the model structure on the method performance is investigated. The possibility of obtaining reliable estimates of coupling is numerically demonstrated, even if the structure of the constructed forecasting model differs from that of the reference system.

The problem of detection and quantitative estimation of directional couplings (mutual influences) between systems from discrete records of their oscillations (time series) arises in different fields of research. This work shows that results of the traditional «Granger causality» approach depend essentially on a sampling interval (a time step). We have revealed the causes and character of the influence of a sampling interval on numerical values of coupling estimates.

The task of detection statistically significant interaction, its direction and delay between time data series of two oscillatory systems in case of close coupling is investigated with nonlinear modeling approach. Numerical experiments on oscillatory model systems with different coupling function variants are used to study main dependences.

The problems of existence and stability of the symmetry-induced nonlinear normal modes in the electric chain of non-linear capacitors, connected to each other with linear inductors (the model described in Physica D238 (2009) 1228) are investigated. For all modes of this type, the upper limit of the stability region (in amplitude of voltage oscillations on capacitors) as a function of the chain cell number were found. Asymptotic formulas were determined at cell number tends to infinity.

We explore the destruction of phase multistability which takes place in an ensemble of period doubling oscillators under the action of long-distance couplings, which appear randomly between the arbitrary cells. The investigation is carried out on the example of a chain of Rossler’s oscillators with periodic boundary conditions, where alongside with local couplings between the elements exist long-range interconnections. The sequence of bifurcations, which accompany increasing of the strength of the global coupling is determined.