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Adilova A. B., Ryskin N. M. Study of synchronization in the system of two delay-coupled gyrotrons using a modified quasilinear model. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2018, vol. 26, iss. 6, pp. 68-81. DOI: 10.18500/0869-6632-2018-26-6–68-81

# Study of synchronization in the system of two delay-coupled gyrotrons using a modified quasilinear model

Topic. The paper is devoted to the study of mutual synchronization of two gyrotrons coupled with delay. As a rule, a theoretical study of synchronization of gyrotrons and other microwave oscillators is usually carried out by numerical simulations using certain well-established models of microwave electronics. Using this approach, it is difficult to provide a fairly complete synchronization pattern, using methods and ideas of nonlinear dynamics. Aim. The aim of the paper is to develop a modified quasilinear model based on the approximation of the electron susceptibility. Methods. The study is based on a bifurcation analysis of the system, which is applicable to this model. A comparison is also made with numerical simulation using the non-stationary theory of a gyrotron with a fixed high-frequency field profile. Results The model proposed in this work allowed us to construct synchronization areas on the plane of parameters of the coupling coefficient – the frequency mismatch for various synchronous modes, the number of which increases with increasing delay time. The model also makes it possible to compute the most important practical parameters (power, efficiency, oscillation frequency). Discussion. An important advantage of the proposed modified quasilinear model is the ability to use modern automated bifurcation analysis packages for studying synchronization modes.

- Landa P.S. Nonlinear Oscillations and Waves in Dynamic Systems, Kluwer, Dordrecht, 1996.
- Pikovsky A., Rosenblum M., Kurths J. Synchronization: A Universal Concept in Nonlinear Science. Cambridge University Press, Cambridge, 2001.
- Kuznetsov A.P., Kuznetsov S.P., Ryskin N.M. Nonlinear Oscillations. Moscow, Fizmatlit, 2005. 292 p. (in Russian).
- York R.A., Compton R.C. Quasi-optical power combining using mutually synchronized oscillator arrays. IEEE Trans. Microwave Theory Tech., 1991, vol. 39, no. 6, pp. 1000–1009.
- Glyavin M.Yu., Kulygin M.L. Theoretical and Experimental Investigation of Auto-Modulation Lasing Regimes in Gyrotrons with Delayed Feedback. Selected Papers of the Contest of Young Scientists. Nizhny Novgorod: IAP RAS, 2001, pp. 16–24 (in Russian).
- Rozental R.M., Ginzburg N.S., Glyavin M.Yu., Sergeev A.S., Zotova I.V. Mutual synchronization of weakly coupled gyrotrons. Phys. Plasmas, 2015, vol. 22, no. 9, 093118.
- Zotova I.V., Ginzburg N.S., Denisov G.G., Rozental R.M., Sergeev A.S. Frequency locking and stabilization regimes in high-power gyrotrons with low-Q resonators. Radiophysics and Quantum Electronics, 2015, vol. 58, no. 9, pp. 759–769.
- Bakunin V.L., Denisov G.G., Novozhilova Yu.V. Zones of frequency locking by an external signal in a multimode gyrotron of a megawatt power level. Radiophysics and Quantum Electronics, 2016, vol. 58, no. 12, pp. 999–1011.
- Yakunina K.A., Kuznetsov A.P., Ryskin N.M. Injection locking of an electronic maser in the hard excitation mode. Phys. Plasmas, 2015, vol. 22, no. 11, 113107.
- Novozhilova Yu.V., Denisov G.G., Glyavin M.Yu., Ryskin N.M., Bakunin V.L., Bogdashov A.A., Melnikova M.M., Fokin A.P. Gyrotron frequency stabilization under the influence of external monochromatic signal or wave reflected from the load: Review. Izvestiya VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, no. 1, pp. 4–34 (in Russian).
- Glyavin M.Yu., Denisov G.G., Kulygin M.L., Melnikova M.M., Novozhilova Yu.V., Ryskin N.M. Gyrotron frequency stabilization by a weak reflected wave. Radiophysics and Quantum Electronics, 2016, vol. 59, no. 9, pp. 673–683.
- Sakamoto K. Progress of high-power-gyrotron development for fusion research. Fusion Sci. Tech., 2007, vol. 52, pp. 145–153.
- Klinshov V.V., Nekorkin V.I. Synchronization of delay-coupled oscillator networks. Physics– Uspekhi, 2013, vol. 56, no. 12, pp. 1217–1229.
- Usacheva S.A., Ryskin N.M. Phase locking of two limit cycle oscillators with delay coupling. Chaos, 2014, vol. 24, no. 2, 023123.
- Adilova A.B., Gerasimova S.A., Ryskin N.M. Bifurcation analyses of mutual synchronization of two oscillators coupled with delay. Nonlinear Dynamics, 2017, vol. 13, no. 1, pp. 3–12 (in Russian).
- Kuznetsov A.P., Stankevich N.V., Turukina L.V. Coupled van der Pol–Duffing oscillators: Phase dynamics and structure of synchronization tongues. Physica D, 2009, vol. 238, no. 14, pp. 1203– 1215.
- http://www.math.pitt.edu/ bard/xpp/xpp.html
- Engelborghs K., Luzyanina T., Roose D. Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Trans. Math. Software, 2002, vol. 28, no. 1, pp. 1–21.
- Nusinovich G.S. Introduction to the Physics of Gyrotrons. Baltimore, London, The Johns Hopkins University Press, 2004.
- Bakunin V.L., Denisov G.G., Zavol’skij N.A., Moiseev M.A. Zones of stable single-mode generation in overmoded gyrotrons. Izvestiya VUZ. Applied Nonlinear Dynamics, 2012, vol. 20, no. 6, pp. 67–81 (in Russian).
- Novozhilova Yu.V., Ryskin N.M., Usacheva S.A. Nonstationary processes in an oscillator with delayed reflection from the load. Technical Physics, 2011, vol. 56, no. 9, pp. 1235–1242.
- Lewis S.M., Nanni E.A., Temkin R.J. Direct machining of low-loss THz waveguide components with an RF choke. IEEE Microw. Wireless Comp. Lett., 2014, vol. 24, no. 12, pp. 842–844.

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