Characteristics of Chaotic Regimes in a Space-distributed Gyroklystron Model with Delayed Feedback

R. M. Rozental ; O. B. Isaeva ; N. S. Ginzburg ; I. V. Zotova ; A. S. Sergeev ; A. G. Rozhnev

Russian journal of nonlinear dynamics, Tome 14 (2018) no. 2, pp. 155-168.

Résumé

Within the framework of the nonstationary model with nonfixed field structure, we investigate the model of a 3-mm band gyroklystron with delayed feedback. It is shown that both chaotic and hyperchaotic generation regimes are possible in this system. The chaotic regime due to a Feigenbaum period-doubling cascade is characterized by one positive Lyapunov exponent. Further transition to hyperchaos is determined by the appearance of another positive exponent in the Lyapunov spectrum. The correlation dimension of the corresponding attractors reaches values of about 3.5.

Détail

Keywords: hyperchaos, Lyapunov exponents
Mots-clés : chaos, gyroklystron.

Bibliographie

[1] Afanas'eva, V. V. and Lazerson, A. G., “Dynamical Chaos in Two-Cavity Klystron-Type Oscillators with Delayed Feedback”, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 3:5 (1995), 88–99 (Russian)

[2] Anishchenko, V. S., Vadivasova, T. E., and Sosnovtseva, O., “Mechanisms of Ergodic Torus Destruction and Appearance of Strange Nonchaotic Attractors”, Phys. Rev. E, 53:5 (1996), 4451–4456 | DOI | MR

[3] Izv. Vyssh. Uchebn. Zaved. Radiofizika, 54:3 (2011), 185–194 (Russian) | DOI

[4] Atino, A., Bonchomme, G., and Pierre, T., “Ionization Waves: From Stability to Chaos and Turbulence”, Eur. Phys. J. D, 19:1 (2002), 79–87 | DOI

[5] Balyakin, A. A. and Ryskin, N. M., “Peculiarities of Calculation of the Lyapunov Exponents Set in Distributed Self-Oscillated Systems with Delayed Feedback”, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 15:6 (2007), 3–21 (Russian) | Zbl

[6] Balyakin, A. A. and Blokhina, E. V., “On the Calculation of the Lyapunov Exponents Set in Distributed Radiophysical Systems”, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 16:2 (2008), 87–110 (Russian) | Zbl

[7] Bezruchko, B. P., Bulgakova, L. V., Kuznetsov, S. P., and Trubetskov, D. I., “Stochastic Self-Oscillations and Instability in a Backward Wave Tube”, Radiotekhnika i Elektronika, 28:6 (1983), 1136–1139 (Russian)

[8] Bezruchko, B. P. and Smirnov, D. A., Extracting Knowledge from Time Series: An Introduction to Nonlinear Empirical Modeling, Springer Series in Synergetics, Springer, Heidelberg, 2010, xxii+405 pp. | DOI | MR | Zbl

[9] Benettin, G., Galgani, L., Giorgilli, A., and Strelcyn, J.-M., “Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems: A Method for Computing All of Them: P. 1: Theory; P. 2: Numerical Application”, Meccanica, 15 (1980), 9–30 | DOI

[10] Blokhina, E. V., Kuznetsov, S. P., and Rozhnev, A. G., “High-Dimensional Chaos in a Gyrotron”, IEEE Trans. Electron Devices, 54:2 (2007), 188–193 | DOI

[11] Bradley, E. and Kantz, H., “Nonlinear Time-Series Analysis Revisited”, Chaos, 25:9 (2015), 097610, 11

[12] Brown, R., Bryant, P., and Abarbanel, H. D. I., “Computing the Lyapunov Spectrum of a Dynamical System from an Observed Time Series”, Phys. Rev. A, 43:6 (1991), 2787–2806 | DOI | MR

[13] Dikhtyar, V. B. and Kislov, V. Ya., “Calculations of Self-Oscillators with External Delayed Feedback by Run Time Method”, Radiotekhnika i Elektronika, 22:10 (1977), 2141–2147 (Russian)

[14] Dmitrieva, T. V., Ryskin, N. M., Titov, V. N., and Shigaev, A. M., “Complex Dynamics of Simple Models of Distributed Electron-Wave Systems”, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 7:6 (1999), 66–81 (Russian)

[15] Eckmann, J.-P. and Ruelle, D., “Ergodic Theory of Chaos and Strange Attractors”, Rev. Modern Phys., 57:3, part 1 (1985), 617–656 | DOI | MR | Zbl

[16] Eckmann, J.-P., Kamphorst, S. O., Ruelle, D., and Gilberto, D., “Lyapunov Exponents from a Time Series”, Phys. Rev. A, 34:6 (1986), 4971–4979 | DOI | MR

[17] Emelyanov, V. V., Girevoy, R. A., Yakovlev, A. V., and Ryskin, N. M., “Time-Domain Particle-in-Cell Modeling of Delayed Feedback Klystron Oscillators”, IEEE Trans. Electron Devices, 61:6 (2014), 1842–1847 | DOI

[18] Ergakov, V. S. and Moiseev, M. A., “Two-Cavity Oscillator with Delayed Feedback”, Radiotekhnika i Elektronika, 31:5 (1986), 962–967 (Russian)

[19] Feudel, U., Kuznetsov, S., and Pikovsky, A., Strange Nonchaotic Attractors: Dynamics between Order and Chaos in Quasiperiodically Forced Systems, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, 56, World Sci., Hackensack, N.J., 2006, xii+213 pp. | MR

[20] Ginelli, F., Poggi, P., Turchi, A., Chaté, H., Livi, R., and Politi, A., “Characterizing Dynamics with Covariant Lyapunov Vectors”, Phys. Rev. Lett., 99:13 (2007), 130601, 4 pp. | DOI

[21] Ginelli, F., Chaté, H., Livi, H., and Politi, A., “Covariant Lyapunov Vectors”, J. Phys. A, 46:25 (2013), 1–25 | DOI | MR

[22] Ginzburg, N. S., Rozental, R. M., Sergeev, A. S., and Zotova, I. V., “Time-Domain Model of Gyroklystrons with Diffraction Power Input and Output”, Phys. Plasmas, 23:3 (2016), 033108, 5 pp. | DOI

[23] Ginzburg, N. S., Sergeev, A. S., Zotova, I. V., and Zheleznov, I. V., “Time-Domain Theory of Gyrotron Traveling Wave Amplifiers Operating at Grazing Incidence”, Phys. Plasmas, 22:1 (2015), 012113, 4 pp. | DOI

[24] Ginzburg, N. S., Zavolsky, N. A., and Nusinovich, G. S., “Theory of Non-Stationary Processes in Gyrotrons with Low Q Resonators”, Int. J. Electronics, 61:6 (1986), 881–894 | DOI

[25] Pis'ma Zh. Tekh. Fiz., 36:2 (2010), 62–69 (Russian) | DOI

[26] Hramov, A. E., Koronovskii, A. A., Maximenko, V. A, and Moskalenko, O. I., “Computation of the Spectrum of Spatial Lyapunov Exponents for the Spatially Extended Beam-Plasma Systems and Electron-Wave Devices”, Phys. Plasmas, 19:8 (2012), 082302, 11 pp. | DOI

[27] Isaeva, O. B., Kuznetsov, A. S., and Kuznetsov, S. P., “Hyperbolic Chaos of Standing Wave Patterns Generated Parametrically by a Modulated Pump Source”, Phys. Rev. E, 87:4 (2013), 040901(R), 4 pp. | DOI

[28] Isaeva, O. B., Kuznetsov, A. S., and Kuznetsov, S. P., “Hyperbolic Chaos in Parametric Oscillations of a String”, Nelin. Dinam., 9:1 (2013), 3–10 (Russian) | DOI

[29] Kanno, K., Uchida, A., and Bunsen, M. A., “Complexity and Bandwidth Enhancement in Unidirectionally Coupled Semiconductor Lasers with Time-Delayed Optical Feedback”, Phys. Rev. E, 93:3 (2016), 032206, 11 pp. | DOI | MR

[30] Kaneko, K., “Doubling of Torus”, Progr. Theoret. Phys., 69:6 (1983), 1806–1810 | DOI | MR | Zbl

[31] Kuptsov, P. V., “Computation of Lyapunov Exponents for Spatially Extended Systems: Advantages and Limitations of Various Numerical Methods”, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 18:5 (2010), 93–112 (Russian) | Zbl

[32] Kuptsov, P. V. and Parlitz, U., “Theory and Computation of Covariant Lyapunov Vectors”, J. Nonlinear Sci., 22:5 (2012), 727–762 | DOI | MR | Zbl

[33] Kuptsov, P. V., “Fast Numerical Test of Hyperbolic Chaos”, Phys. Rev. E, 85:1 (2012), 015203(R), 4 pp. | DOI

[34] Kuptsov, P. V. and Kuznetsov, S. P., “Numerical Test for Hyperbolicity of Chaotic Dynamics in Time-Delay Systems”, Phys. Rev. E, 94:1 (2016), 010201(R), 7 pp. | DOI | MR

[35] Izv. Vyssh. Uchebn. Zaved. Radiofizika, 25:12 (1982), 1410–1428 (Russian) | DOI

[36] Kuznetsov, S. P., Dynamical Chaos, 2nd ed., Fizmatlit, Moscow, 2006 (Russian)

[37] Izv. Vyssh. Uchebn. Zaved. Radiofizika, 47:5–6 (2004), 383–398 (Russian) | DOI

[38] Nusinovich, G. S., Introduction to the Physics of Gyrotrons, J. Hopkins Univ. Press, Baltimore, 2004, 341 pp.

[39] Pedersen, T. S., Mechelsen, P. K., and Rasmussen, J. J., “Lyapunov Exponents and Particle Dispersion in Drift Wave Turbulence”, Phys. Plasmas, 3:8 (1996), 2939–2950 | DOI | MR

[40] Persson, N. and Nordman, H., “Low-Dimensional Chaotic Attractors in Drift Wave Turbulence”, Phys. Rev. Lett., 67:24 (1991), 3396–3399 | DOI

[41] Ruelle, D., “Ergodic Theory of Differentiable Dynamical Systems”, Inst. Hautes Études Sci. Publ. Math., 1979, no. 50, 27–58 | DOI | MR | Zbl

[42] Sano, M. and Sawada, Y., “Measurement of the Lyapunov Spectrum from a Chaotic Time Series”, Phys. Rev. Lett., 55:10 (1985), 1082–1085 | DOI | MR

[43] Sauer, T. D., Tempkin, J. A., and Yorke, J. A., “Spurious Lyapunov Exponents in Attractor Reconstruction”, Phys. Rev. Lett., 81:20 (1998), 4341–4344 | DOI

[44] Thumm, M., State-of-the-Art of High Power Gyro-Devices and Free Electron Masers, KIT Sci. Rep., 7735, KIT Sci., Karlsruhe, 2017, 196 pp.

[45] Tolkachev, A. A., Levitan, B. A., Solovjev, G. K., Veytsel, V. V., and Farber, V. E., “A Megawatt Power Millimeter-Wave Phased-Array Radar”, IEEE Aerosp. Electron. Syst. Mag., 15:7 (2000), 5–31 | DOI

[46] Izv. Vyssh. Uchebn. Zaved. Radiofizika, 12:8 (1969), 1236–1244 (Russian) | DOI

[47] Wolf, A., Swift, J. B., Swinney, H. L., and Vastano, J. A., “Determining Lyapunov Exponents from a Time Series”, Phys. D, 16:3 (1985), 285–317 | DOI | MR | Zbl

[48] Yang, H., Radons, G., and Kantz, H., “Covariant Lyapunov Vectors from Reconstructed Dynamics: The Geometry behind True and Spurious Lyapunov Exponents”, Phys. Rev. Lett., 109:24 (2012), 244101, 4 pp. | DOI

[49] Izv. Vyssh. Uchebn. Zaved. Radiofizika, 55:5 (2012), 341–350 (Russian) | DOI

[50] Zeng, X., Eykholt, R., and Pielke, R. A., “Estimating the Lyapunov Exponent Spectrum from Short Time Series of Low Precision”, Phys. Rev. Lett., 66:25 (1991), 3229–3232 | DOI