Synchronization of oscillators with hard excitation coupled with delay. Part 2. Amplitude-phase approximation

A. B. Adilova ; N. M. Ryskin

Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 32 (2024) no. 5, pp. 574-588.

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. A bifurcation analysis of the mutual synchronization of two generators with hard excitation in the amplitude-phase approximation is carried out. The results of the bifurcation analysis are compared with the results of numerical simulation of the system of differential equations with delay. Results. A complete bifurcation pattern of mutual synchronization on the plane "frequency mismatch - coupling parameter" is presented. In the case of small mismatch and weak coupling, the fixed points, which correspond to modes with dominance of one of the oscillators, merge with saddle fixed points and disappear when the coupling parameter increases. In the case of large mismatch, one of these points either vanishes or loses stability as a result of a subcritical Andronov-Hopf bifurcation. The other of these points remains stable at any values of the coupling parameter, and the oscillation amplitudes of both oscillators gradually equalize and the phase difference tends to zero, i.e., the oscillation mode with dominance of one of the oscillators gradually transforms into the in-phase synchronization mode. It has been found that with an increase in the coupling parameter, a transformation of the basin of attraction of a stable zero fixed point occurs. As a result of this transformation, if at the initial moment of time the oscillations of the generators are close to antiphase, the oscillations decay at any initial amplitudes. Conclusion. The synchronization pattern in the system of delay-coupled oscillators with hard excitation has been studied. It was discovered that in addition to mutual synchronization modes with approximately equal oscillation amplitudes, stationary modes with suppression of oscillations of one generator by another are also possible. The bifurcation mechanisms of the appearance and disappearance of multistability in the system have been examined.

Détail

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

Bibliographie

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

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

[3] 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

[4] Korolev V. I., Postnikov L. V., “K teorii sinkhronizatsii generatora avtokolebanii. I”, Izvestiya vuzov. Radiofizika, 12:3 (1969), 406–414

[5] Kuznetsov A. P., Milovanov S. V., “Sinkhronizatsiya v sisteme s bifurkatsiei sliyaniya ustoichivogo i neustoichivogo predelnykh tsiklov”, Izvestiya vuzov. Prikladnaya nelineinaya dinamika, 11:4/5 (2003), 16–30 | DOI | Zbl

[6] Milovanov S. V., Sinkhronizatsiya sistem s sosuschestvuyuschimi ustoichivym i neustoichivym predelnymi tsiklami i bifurkatsiei ikh sliyaniya i ischeznoveniya, diss. ... k.f.-m.n., SGU, Saratov, 2005, 209 pp.

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

[8] Grigoreva N. V., Ryskin N. M., “Issledovanie sinkhronizatsii girotrona v rezhime zhestkogo vozbuzhdeniya na osnove modifitsirovannoi kvazilineinoi modeli”, Izvestiya vuzov. Radiofizika, 65:5/6 (2022) | DOI

[9] Adilova A. B., Ryskin N. M., “Cinkhronizatsiya generatorov s zhestkim vozbuzhdeniem, svyazannykh s zaderzhkoi. Chast 1. Fazovoe priblizhenie”, Izvestiya vuzov. Prikladnaya nelineinaya dinamika, 32:1 (2024), 42–56 | DOI

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

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

[12] Izhikevich E. M., “Phase models with explicit time delays”, Physical Review E, 58:1 (1998), 905–908 | DOI

[13] Glyzin S. D., “Dinamicheskie svoistva prosteishikh konechnoraznostnykh approksimatsii kraevoi zadachi «reaktsiya-diffuziya»”, Differentsialnye uravneniya, 33:6 (1997), 805–811 | MR | Zbl

[14] Jessop M. R., Li W., Armour A. D., “Phase synchronization in coupled bistable oscillators”, Physical Review Research, 2:1 (2020), 013233 | DOI

[15] Burić N., Grozdanović I., Vasović N., “Excitable systems with internal and coupling delays”, Chaos, Solitons Fractals, 36:4 (2008), 853–861 | DOI | MR

[16] Shilnikov L. P., “O nekotorykh sluchayakh rozhdeniya periodicheskikh dvizhenii iz osobykh traektorii”, Matematicheskii sbornik, 61(103):4 (1963), 443–466 | MR | Zbl

[17] XPPAUT, [Elektronnyi resurs] http://www.math.pitt.edu/~bard/xpp/xpp.html

[18] Peregorodova E. N., Ryskin N. M., Usacheva S. A., “Sinkhronizatsiya sistemy dvukh konkuriruyuschikh mod vneshnim garmonicheskim signalom”, Izvestiya vuzov. Prikladnaya nelineinaya dinamika, 19:3 (2011), 154–170 | DOI | Zbl

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