651 ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2017, Vol. 125, No. 4, pp. 651–662. © Pleiades Publishing, Inc., 2017. Original Russian Text © V.A. Buts, V.V. Kuz’min, A.P. Tolstoluzhsky, 2017, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2017, Vol. 152, No. 4, pp. 767–780. Features of the Dynamics of Particles and Fields at Cyclotron Resonances V. A. Buts a,b,c, *, V. V. Kuz’min a,c , and A. P. Tolstoluzhsky a a National Science Center Kharkiv Institute of Physics and Technology, Kharkiv, 61108 Ukraine b Institute of Radioastronomy, National Academy of Sciences of Ukraine, Kharkiv, 61002 Ukraine c Kharkiv National University, Kharkiv, 61022 Ukraine *e-mail: vbuts@kipt.kharkov.ua Received March 20, 2017 Abstract—Some features of the dynamics of particles and fields at cyclotron resonances have been discussed with the focus on chaotic dynamical regimes. It has been shown that the known criterion of the transition of the regular dynamics of particles to chaotic dynamics at cyclotron resonances sometimes describes this tran- sition incorrectly. The reason for such a feature of the criterion has been revealed. The anomalous sensitivity of the dynamics of particles to external fluctuations at autoresonance has been analyzed. A theory of exci- tation of electromagnetic waves by a beam of phased oscillators under the conditions of isolated nonlinear cyclotron resonance has been developed. It has been shown that the chaotic dynamical regime is due to the periodic change in the structure of the phase portrait of particles in the wave field. It has been shown that higher moments can play a more significant role than lower moments in almost all chaotic dynamical regimes at cyclotron resonances. In this case, the known kinetic diffusion equations should be generalized with the inclusion of these higher moments. DOI: 10.1134/S1063776117090126 1. INTRODUCTION Electron cyclotron resonance (ECR) is one of the most known, studied, and widely used resonances. Electron cyclotron resonance is widely used in elec- tronics (see, e.g., [1]) and in many schemes of high- frequency heating of a high-temperature plasma in controlled nuclear fusion facilities and in facilities used as sources of neutrons (see, e.g., [2–6]). The the- ory and properties of this resonance are well studied. It is known that this resonance makes it possible to cre- ate the most efficient schemes for energy exchange between electromagnetic waves and beams of charged particles. Mechanisms of such interaction are also well studied. Any new features of ECR were not expected. Features of experimental setups, where the interaction of electromagnetic waves with charged particles occurs, should be taken into account in each particular experiment with ECR. In particular, the authors of [3–5] analyzed in detail schemes of ECR heating of the plasma for particular setups. In this case, it is nec- essary to take into account inhomogeneities of the plasma, inhomogeneities of an external magnetic field, possible spatial regions of trap of an electromag- netic field, and many other features of ECR in partic- ular experimental setups. This is a rather difficult problem. All new features of ECR seem to be associ- ated with the requirements of real experiments. How- ever, it will be shown below that ECR has quite general features that have not yet been studied. These features primarily concern chaotic dynamical regimes. The study of ECR involves the most significant problem of conditions for the regular and chaotic dynamics of particles in these resonances. Criteria for transition from the regular dynamics of particles at ECR to chaotic dynamics were obtained in [7, 8]. These criteria were analyzed numerically in many works (see, e.g., [7–9]). Furthermore, they were con- firmed in experiments [10, 11]. Consequently, the problem of conditions for the regular and chaotic dynamics of particles at ECR seems to be completely solved. However, some our numerical studies of ECR, as well as numerous studies of autoresonance (see [6, 12] and references therein), show that the dynamics of particles in some important cases does not satisfy known criteria. In particular, studies of autoresonance indicate that a high efficiency of energy exchange between charged particles and waves is not reached in experiments [12]. In this work, we try to resolve some appearing contradictions and to reveal some new fea- tures of the dynamics of particles and fields at ECR. The paper is organized as follows. Known condi- tions for chaotic dynamical regimes at cyclotron reso- nances are discussed in Section 2. Contradictions between numerical results and analytical criteria are ELECTRONIC PROPERTIES OF SOLID