In this paper, the self-consistent model was considered, consisting of a system of oscillators, the coupling between them was assumed to be integral (due to the fields formed as a result of their co-radiation). With the help of this model, the features of synchronization by waves of finite amplitude of a system of oscillators were refined, the initial phase values of which are random. The effect of nonlinearity, in particular, due to the change in the mass of the oscillator due to relativistic effects, was taken into account. It was shown that the nonlinearity does not violate the nature of the energy exchange between the wave and the oscillator system, leading only to a slight decrease in the efficiency of such an exchange. KEYWORDS: oscillator, nonlinearity, synchronization, energy exchange пл. Свободы 4, 61022, Харьков, Украина В данной работе была рассмотрена самосогласованная модель, состоящая из системы осцилляторов, связь между которыми предполагалась интегральной (за счет полей, формируемых в результате их совместного излучения). С помощью данной модели были уточнены особенности синхронизации волнами конечной амплитуды системы осцилляторов, начальные значения фаз которых являются случайными. Был проведен учет влияния нелинейности, в частности, обусловленной изменением массы осциллятора за счет релятивистских эффектов. Было показано, что нелинейность не нарушает характер обмена энергией между волной и системой осцилляторов, приводя лишь к небольшому снижению эффективности такого обмена. КЛЮЧЕВЫЕ СЛОВА: осциллятор, нелинейность, синхронизация, обмен энергией Synchronization of systems of oscillators which drew the attention of C. Huygens as early as in the seventeenth century, was discussed by many researchers [1,2]. The influence of external periodic force is able to adjust the parameters of the oscillator and its phase. Therefore, synchronization is often referred to as "locking" of frequency and/or phase [3,4].The most efficient synchronization of a large number of oscillators occurs under the influence of a strong external force or under the impact of already synchronized oscillators.The coupling between the oscillators can be local due to their mutual non-linear influence or have integral nature due to the fields generated by their joint radiation. Of great interest are the various parametric mechanisms affecting the efficiency of interaction of oscillators [5,6], although in this work, we neglect their direct interaction. Instead, we focus on the case when independent oscillators interact via the common field (generated by them and influencing them).An important factor of interest to this case is the apparent self-consistency of the described systems and the great practical importance of such problems. The considered models are able to clarify some useful properties of oscillators' synchronization by the finite-amplitude waves when initial phases of oscillators are random. Also, we studied nonlinear effects, particularly caused by the change of the oscillator's mass due to the relativity effects.So, ...
In the article, for limited system conditions that form the spatial structure of the field, the attenuation processes of wave packets of finite amplitude are considered. The line width of the wave field may be the result of the dissipative processes (in a quantum system it is inverse of the lifetime of energy levels) or the result of reactive processes (in classical waveguide systems this is the spectral width of the packet). In the case of filling the waveguide with an active two-level medium, a description is possible using a quasiclassical model of the interaction of the field and particles. In this case, the quantum-mechanical description of the medium is combined with the classical representation of the field. Here, the Rabi frequency plays an important role, which determines the probabilities of induced radiation or absorption of field quanta and the oscillatory change in population inversion (nutation). Depending on the relationship between the Rabi frequency and the line width of the wave packet, the process can change the nature of the field behavior. In strong fields or with a significant population inversion, the line width can be neglected, while the field energy density is quite high. In this case, one should expect noticeable nutations of population inversions with different frequencies corresponding to the local Rabi frequency in different regions of the waveguide, the interference of which will determine the oscillatory behavior of the wave field. At a low level of electric field intensity or a slight population inversion, the mode of changing the field amplitude becomes monotonic. Plasma field damping (Landau damping) is considered. The role of population inversion is assumed by a quantity proportional to the derivative with respect to velocity of the electron distribution function. If the spectral width of the packet is small, the process of wave attenuation acquires a characteristic oscillatory form due to the exchange of energy between the wave and the plasma electrons captured by its field. The attenuation of wide packets is almost monotonic with the formation of a characteristic “plateau” in the vicinity of the phase velocity of the wave on the electron velocity distribution function.
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