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Purpose of this work is the research of the dynamical processes and in particular the phenomenon of the synchronization in an ensemble of coupled chaotic economic oscillators. Methods. The research methods are the qualitative and numerical methods of the theory of nonlinear dynamical systems and the theory of the bifurcations. Results. The nonlinear model of economic oscillator as the system of automatic control are considered. Such kind of general economic models are unsuitable for getting some concrete economic estimations and recommendations. But such kind models are very useful for a development the theory of the economic cycles, theory of the generation, interactions, synchronization of the cycles and so on. Our numerical experiments demonstrated a good enough qualitative similarity of an chaotic economic oscillations in our model and real economic cycles. The phenomen of the synchronization of the chaotic oscillations in the ensemble of coupled economic oscillators are considered, however the accuracy of the synchronization depends with couplings essentially.

Purpose of the work is to research of the world science publications dynamics on the synchronization. Methods. The research methods are the statistical methods of data processing. Results. The emphasis in the study of synchronization over the past twenty years has shifted from physical and technical sciences to neuroscience with Asian countries domination.

Purpose of this work is to study a computational model of working memory formation based on spiking neural network with plastic connections and to study the capacity of working memory depending on the time scales of synaptic facilitation and depression and the background excitation of the network. Methods. The model imitates working memory formation within synaptic theory: memorized items are stored in form of short-term potentiated connections in selective population but not in form of persistent activity. Integrate-And-Fire neuron model in excitable mode are used as network elements. Connections between excitatory neurons demonstrates the effect of short-term plasticity. Results. It is shown that the working memory capacity increases as calcium recovery time parameter grow up or the capacity increases with neurotransmitter recovery time parameter becomes lower. Working memory capacity is found to decrease to zero with decrease of the background excitation as a result of lower values of both the mean and the variance of the external noise. Conclusion. Working memory capacity was studied as a function of time scales of synaptic facilitation and depression and background excitation of the network. Estimated working memory capacity is shown to be possibly larger than classical experimental estimations of four items. But capacity strongly depends on intrinsic parameters of neural networks.

Purpose of this work is to develop a reconstruction technique for the equations of a phase-locked loop system under periodic external driving from a scalar time series of one variable. Methods. Instead of the original model, we reconstructed a time-integrated model. So, since it is not necessary to evaluate the second derivative of the observable numerically, the method sensitivity to observation noise has significantly decreased. The external periodic driving is approximated with a trigonometric polynomial of time, the antiderivative of which is also a trigonometric polynomial. The assumption about continuity of an unknown nonlinear function is used to construct the target function for optimization. Results. It is shown that the proposed approach gives a significant advantage over the previously developed approach to the reconstruction of non-integrated equations, allowing to achieve acceptable parameter estimates with measurement noise being about 10% of the RMS deviation of the signal even in the presence of external driving. Conclusion. The described approach significantly extends the possibilities of reconstruction of phase-locked loop systems, allowing systems to be reconstructed under arbitrary periodic driving and at the same time significantly increasing noise resistance.

The purpose of this work is to research the phenomena of the self-oscillation and the synchronization for the model of business and financial oscillator, which presented as the system of automatic control. Methods. The research methods are the qualitative and numerical methods of the theory of nonlinear dynamical systems and the theory of the bifurcations. Results. This work presents the model of business and financial oscillators as the phase-controlled oscillator and as the frequency-controlled oscillator. The phenomena of the self-oscillation of regular and chaotic oscillations in this model and the synchronization of such oscillations are considered.

Abstract. The purpose of this work is to study the dynamics of phase-controlled oscillator in excitable mode under rectangular pulse train forcing. The excitable system is a system having a stable equilibrium and a large amplitude periodic pseudo-orbit passing near the equilibrium. Methods. In this paper, the dynamics of the generator under periodic or Poisson rectangular pulse train stimulation is studied by numerical simulation. Different values are proposed to characterize statistics of the oscillator’s responses on different number of stimulating pulses. Results. The role of amplitude and period of external forcing on generator excitation is studied. Relative frequency of the oscillator responses depends on the amplitude of stimulating pulses. Phase-controlled oscillator’s responses are synchronized with different rational relations to number of stimulating pulses depending on the amplitude of the pulses. The inter-pulse intervals of the oscillator are not completely rational to inter-pulse intervals of stimulating train in contrast to relative frequency of the oscillator responses. The Poisson pulse train stimulation gives nearly the same statistics of the oscillator’s responses as periodic forcing. Conclusion. Detailed study of the dynamics of phase-controlled oscillator in excitable mode under rectangular pulse train forcing is conducted. Different ways of the oscillator’s responses characterization shows the strong dependence on the amplitude of stimulating pulses but much weaker dependence on inter-pulse intervals of stimulating train. The most informative characteristics is a ratio between inter-pulse intervals of oscillator responses and stimulating train.