Synchronization of oscillators with hard excitation coupled with delay. Part 1. Phase approximation

A. B. Adilova ; N. M. Ryskin

Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 32 (2024) no. 1, pp. 42-56.

Résumé

Aim of this work is to develop the theory of mutual synchronization of two oscillators with hard excitation associated with a delay. Taking into account the delay of a coupling signal is necessary, in particular, when analyzing synchronization at microwave frequencies, when the distance between the oscillators is large compared to the wavelength. Methods. Theoretical analysis is carried out under the assumption that the delay time is small compared to the characteristic time for the oscillations. The phase approximation is used when the frequency mismatch and the coupling parameter are considered small. Results. Taking into account the change in oscillation amplitudes up to first-order terms in the coupling parameter, a generalized Adler equation for the phase difference of the oscillators is obtained, which takes into account the combined type of the coupling (dissipative and conservative coupling) and non-isochronism. The conditions for saddle-node bifurcations are found and the stability of various fixed points of the system is analyzed. The boundaries of the domains of in-phase and anti-phase synchronization are plotted on the plane of the parameters "frequency mismatch - coupling parameter". Conclusion. It is shown that, depending on the control parameters (non-isochronism parameter, excitation parameter, phase advance of the coupling signal), the system exhibits behavior typical of either dissipative or conservative coupling. The obtained formulas allow for trace the transition from one type of coupling to another when varying the control parameters.

Détail

Keywords: coupled generators, self-oscillating systems with hard excitation, synchronization, delay, phase approximation, generalized Adler equation

Bibliographie

[1] Rabinovich M. I., Trubetskov D. I., Vvedenie v teoriyu kolebanii i voln, Nauka, M., 1984, 432 pp.

[2] Pikovskii A., Rozenblyum M., Kurts Yu., Sinkhronizatsiya. Fundamentalnoe nelineinoe yavlenie, Tekhnosfera, M., 2003, 496 pp.

[3] Kuznetsov A. P., Kuznetsov S. P., Ryskin N. M., Nelineinye kolebaniya, Fizmatlit, M., 2005, 292 pp.

[4] Balanov A., Janson N., Postnov D., Sosnovtseva O., Synchronization: From Simple to Complex, Springer, Berlin, 2009, 426 pp. | DOI | Zbl

[5] Kuznetsov A. P., Emelyanova Yu. P., Sataev I. R., Tyuryukina L. V., Sinkhronizatsiya v zadachakh, Izdatelskii tsentr «Nauka», Saratov, 2010, 256 pp.

[6] Zhang J., Zhang D., Fan Y., He J., Ge X., Zhang X., Ju J., Xun T., “Progress in narrowband high-power microwave sources”, Phys. Plasmas, 27:1 (2020), 010501 | DOI | MR

[7] Benford J., Sze H., Woo W., Smith R. R., Harteneck B., “Phase locking of relativistic magnetrons”, Phys. Rev. Lett., 62:8 (1989), 969–971 | DOI

[8] Cruz E. J., Hoff B. W., Pengvanich P., Lau Y. Y., Gilgenbach R. M., Luginsland J. W., “Experiments on peer-to-peer locking of magnetrons”, Appl. Phys. Lett., 95:19 (2009), 191503 | DOI

[9] Sze H., Price D., Harteneck B., “Phase locking of two strongly coupled vircators”, J. Appl. Phys., 67:5 (1990), 2278–2282 | DOI

[10] Woo W., Benford J., Fittinghoff D., Harteneck B., Price D., Smith R., Sze H., “Phase locking of high-power microwave oscillators”, J. Appl. Phys., 65:2 (1989), 861–866 | DOI

[11] Levine J. S., Aiello N., Benford J., Harteneck B., “Design and operation of a module of phase-locked relativistic magnetrons”, J. Appl. Phys., 70:5 (1991), 2838–2848 | DOI

[12] Adilova A. B., Ryskin N. M., “Issledovanie sinkhronizatsii v sisteme dvukh girotronov s zapazdyvaniem v kanale svyazi na osnove modifitsirovannoi kvazilineinoi modeli”, Izvestiya vuzov. PND, 26:6 (2018), 68–81 | DOI

[13] Adilova A. B., Ryskin N. M., “Vliyanie zapazdyvaniya na vzaimnuyu sinkhronizatsiyu dvukh svyazannykh girotronov”, Izvestiya vuzov. Radiofizika, 63:9–10 (2020), 781–795

[14] Thumm M. K. A., Denisov G. G., Sakamoto K., Tran M. Q., “High-power gyrotrons for electron cyclotron heating and current drive”, Nucl. Fusion, 59:7 (2019), 073001 | DOI

[15] Ivanchenko M. V., Osipov G. V., Shalfeev V. D., Kurths J., “Synchronization of two non-scalar-coupled limit-cycle oscillators”, Physica D, 189:1–2 (2004), 8–30 | DOI | MR | Zbl

[16] Kuznetsov A. P., Stankevich N. V., Turukina L. V., “Coupled van der Pol–Duffing oscillators: Phase dynamics and structure of synchronization tongues”, Physica D, 238:14 (2009), 1203–1215 | DOI | MR | Zbl

[17] Usacheva S. A., Ryskin N. M., “Phase locking of two limit cycle oscillators with delay coupling”, Chaos, 24:2 (2014), 023123 | DOI | MR | Zbl

[18] Adilova A. B., Gerasimova S. A., Ryskin N. M., “Bifurkatsionnyi analiz vzaimnoi sinkhronizatsii dvukh generatorov s zapazdyvaniem v tsepi svyazi”, Nelineinaya dinamika, 13:1 (2017), 3–12 | DOI | MR | Zbl

[19] Adilova A. B., Balakin M. I., Gerasimova S. A., Ryskin N. M., “Bifurcation analysis of multistability of synchronous states in the system of two delay-coupled oscillators”, Chaos, 31:11 (2021), 113103 | DOI | MR

[20] Nusinovich G. S., Introduction to the Physics of Gyrotrons, Johns Hopkins University Press, Baltimore, 2004, 352 pp. | DOI

[21] Yakunina K. A., Kuznetsov A. P., Ryskin N. M., “Injection locking of an electronic maser in the hard excitation mode”, Phys. Plasmas, 22:11 (2015), 113107 | DOI

[22] Kuznetsov A. P., Sataev I. R., Trubetskov D. I., Seliverstova E. S., “Udivitelnyi Robert Adler: lampa Adlera, uravnenie Adlera i mnogoe drugoe”, Izvestiya vuzov. PND, 23:3 (2015), 3–26 | DOI | MR | Zbl