For citation:
Ryskin N. M., Khavroshin O. S. Controlling chaos in Ikeda system. Symplified discrete map model. Izvestiya VUZ. Applied Nonlinear Dynamics, 2009, vol. 17, iss. 2, pp. 66-86. DOI: 10.18500/0869-6632-2009-17-2-66-86
Controlling chaos in Ikeda system. Symplified discrete map model
Method of controlling chaos in a ring cavity containing a media with cubic phase nonlinearity (Ikeda system) is considered. The proposed method is based on introduction of an additional feedback loop with parameters chosen so that the fundamental frequency components after passing through different feedback loops appear in phase, while the most unstable sidebands appear in antiphase, thus suppressing each other. In the weak dispersion limit a discrete map is derived that is a modification of the well-known Ikeda map. The results of analytic investigation and numerical simulation of this map in a broad range of parameters are presented. It is shown that the suggested method allows suppression of self-modulation and stabilization of periodic regimes.
- Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys. Rev. Lett. 1990;64(11):1196–1199. DOI: 10.1103/PhysRevLett.64.1196.
- Pyragas K. Continuous control of chaos by self-controlling feedback. Phys. Lett. A. 1992;170:421–428.
- Just W, Benner H, Schoell E. Control of chaos by time-delayed feedback: A survey of theoretical and experimental aspects. In: B. Kramer (Ed.), Advances in Solid State Physics. Vol. 43. Berlin: Springer; 2003. 589–603 p.
- Hövel P, Schöll E. Control of unstable steady states by time-delayed feedback methods. Phys Rev E Stat Nonlin Soft Matter Phys. 2005;72(4):046203. DOI: 10.1103/PhysRevE.72.046203.
- Dolov AM, Kuznetsov SP. Chaos-Controlling Technique for Suppressing Self-Modulation in Backward-Wave Tubes. Technical Physics. 2003;48(8):1074–1077. DOI: 10.1134/1.1607485.
- Ikeda K, Daido H, Akimoto O. Optical turbulence: chaotic behavior of transmitted light from a ring cavity. Phys. Rev. Lett. 1980;45(9):709–712. DOI: 10.1103/PHYSREVLETT.45.709.
- Izmailov IV, Lyachin AV, Poizner BN. Deterministic chaos in models of a ring nonlinear interferometer. Tomsk: TSU Publish; 2007. 252 p. (In Russian).
- Rozanov NN. Optical bistability and hysteresis in distributed nonlinear systems. Moscow: Nauka, Fizmatlit; 1997. 333 p. (In Russian).
- Dodd RK, Eilbeck JC, Gibbon JD, Morris HS. Solitons and Nonlinear Wave Equations. London: Academic Press; 1984.
- Ryskin NM, Trubetskov DI. Nonlinear waves. Moscow: Nauka; Fizmatlit; 2000. 268 p. (In Russian).
- Ostrovsky LA, Potapov AI. Introduction to the theory of modulated waves. Moscow: Fizmatlit; 2003. 398 p. (In Russian).
- Balyakin AA, Ryskin NM, Khavroshin OS. Nonlinear dynamics of modulation instability in distributed resonators under external harmonic driving. Radiophysics and Quantum Electronics. 2007;50(9):726–744. DOI: 10.1007/s11141-007-0064-2.
- Kuznetsov SP. Dynamic chaos. Moscow: Fizmatlit; 2001. 296 p. (In Russian).
- Emel'yanov VV, Ryskin NM, Khavroshin OS. Suppression of self-modulation in the self-oscillator with delayed feedback using the method for chaos control. Journal of Communications Technology and Electronics. 2009;54(6):685–691. DOI: 10.1134/S1064226909060102.
- Ryskin NM, Khavroshin OS. Suppressing self-modulation instability in a delayed feedback traveling wave tube oscillator using controlling chaos technique. IEEE Trans. Electron Devices. 2008;55(2):662–667. DOI: 10.1109/TED.2007.912366.
- Kuznetsov AP, Tyuryukina LV. Dynamic systems of different classes as models of a nonlinear oscillator with impulse impact. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(2):31–42.
- Valee R, Delisle C, Chrostowski J. Noise versus chaos in acousto-optic bistability. Phys. Rev. A. 1984;30(1):336–338.
- Kuznetsov AP, Savin AV, Savin DV. Conservative and dissipative dynamics of Ikeda map. Izvestiya VUZ. Applied Nonlinear Dynamics. 2006;14(2):94–106. DOI: 10.18500/0869-6632-2006-14-2-94-106.
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