ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Balyakin A. A. Investigation of chaotic dynamics of a nonlinear ring cavity under two-frequency external driving. Izvestiya VUZ. Applied Nonlinear Dynamics, 2003, vol. 11, iss. 4, pp. 3-15. DOI: 10.18500/0869-6632-2003-11-4-3-15

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Investigation of chaotic dynamics of a nonlinear ring cavity under two-frequency external driving

Balyakin Artem Aleksandrovich, Saratov State University

Complex dynamics of a nonlinear ring cavity filled by medium with cubic phase nonlinearity under multi-frequency driving is considered. System оf coupled Ikeda maps to describe the dynamics of spectral components was derived. Regimes of steady-state oscillations and their stability conditions are analyzed. The results оf numerical simulation of transition to chaos in the case of two-frequency driving are presented.  

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The author is grateful to N.M. Ryskin for his useful advice and discussion results of the work. This work was supported by grants CRDF (Award No. REC-006 ), Russian Foundation for Basic Research (Project No. 03-02-06257).
  1. Landa PS. Nonlinear Oscillations and Waves in Dynamical Systems. Dordrecht: Springer; 1996. 544 p. DOI: 10.1007/978-94-015-8763-1.
  2. Neimark YI, Landa PS. Stochastic and Chaotic Oscillations. Dordrecht: Springer; 1992. 500 p. DOI: 10.1007/978-94-011-2596-3.
  3. Kuznetsov SP. Dynamic Chaos. Moscow: Fizmatlit; 2001. 296 p. (in Russian).
  4. Akhmanov SA, Vorontsov MA. Instabilities and structures in coherent nonlinear optical systems covered by two-dimensional feedback. In: Nonlinear Waves: Dynamics and Evolution. Digest of Articles. Moscow: Nauka; 1989. Р. 228–237.
  5. Ikeda K. Multiple-valued stationary state and its instability of the transmitted light by а ring cavity system. Opt. Comm. 1979;30(2):257–261. DOI: 10.1016/0030-4018(79)90090-7.
  6. Ikeda K, Daido H, Akimoto О. Optical turbulence: chaotic behavior of transmitted light from а ring cavity. Phys. Rev. Lett. 1980;45(9):709–712. DOI: 10.1103/PhysRevLett.45.709.
  7. Ikeda K, Akimoto О. Instability leading to periodic and chaotic self-pulsations in а bistable optical cavity. Phys. Rev. Lett. 1982;48(9):617–620. DOI: 10.1103/PhysRevLett.48.617.
  8. Nakatsuka H, Asaka S, Itoh H, lkeda K, Matsuoka M. Observation of bifurcation to chaos in all-optical bistable system. Phys. Rev. Lett. 1983;50(2):109–112. DOI: 10.1103/PhysRevLett.50.109.
  9. Moloney JV, Newell AC. Nonlinear optics. Physica D. 1990;44(1-2):1-37. DOI: 10.1016/0167-2789(90)90045-Q.
  10. Izmailov IV, Kalaida BT, Magazinnikov AL, Poizner BH. Bifurcations in a point model of a ring interferometer with delay and field rotation. Izvestiya VUZ. Applied Nonlinear Dynamics. 1999;7(5):47–59 (in Russian).
  11. Izmailov NB, Magazinnikov AL, Poizner BN. Modeling of processes in a ring interferometer with nonlinearity, delay and diffusion under non-monochromatic radiation. News of Universities. Physics. 2000;(2):29–35 (in Russian).
  12. Chesnokov SS, Rybak АА. Spatiotemporal chaotic behavior оf time-delayed nonlinear optical systems. Laser Physics. 2000;10(5):1061–1068.
  13. Rozanov NN. Optical Bistability and Hysteresis in Distributed Nonlinear Systems. Moscow: Nauka; 1997. 336 p. (in Russian).
  14. Akhmanov SA, Vorontsov MA, editors. New Physical Principles of Optical Information Processing. Digest of Articles. Moscow: Nauka; 1990. 398 p. (in Russian).
  15. Izmailov IV. Options for implementing a nonlinear optical information security device. Optical Journal. 2002;69(7):62–67 (in Russian).
  16. Bogatov NA, Gitlin MS. Nonlinear microwave quasi-optics. Bulletin of the Russian Academy of Sciences: Physics. 1999;63(12):2340–2349 (in Russian).
  17. Balyakin AA, Ryskin NM. Transition K to chaos in a ring nonlinear resonator under excitation by an external multifrequency signal. Bulletin of the Russian Academy of Sciences: Physics. 2001;65(12):1741–1744 (in Russian).
  18. Balyakin AA, Ryskin NM. Chaotic oscillations in nonlinear spatially-extended resonators. Nonlinear Phenomena in Complex Systems. 2001;4(4):358–363.
  19. Pliszka P, Banerjee PP. Analysis of multifrequency dispersive optical bistability and switching in nonlinear ring cavities with large medium-response times. Phys. Rev. A. 1992;46(1):507–517. 10.1103/PhysRevA.46.507.
  20. Zakharov VE. The Hamiltonian Formalism for waves in nonlinear media having dispersion. Radiophys. Quantum Electron. 1974;17(4):326–343. DOI: 10.1007/BF01036794.
  21. Zakharov VE, Kuznetsov EA. Hamiltonian formalism for nonlinear waves. Phys. Usp. 1997;40(11):1087–1116. DOI: 10.1070/PU1997v040n11ABEH000304.
  22. Ryskin NM. Coupled nonlinear Schrödinger equations for multifrequency wave packets in a dispersive nonlinear medium. JETP. 1994;79(5):833–835.
  23. Agrawal G. Nonlinear Fiber Optics. San Diego, CA: Academic Press; 1989.
  24. Kuznetsov AP, Turukina LV, Mosekilde Е. Dynamical systems оf different classes аs models оf the kicked nonlinear oscillator. Internat. Journ. Bifurcation and Chaos. 2001;11(4):1065–1077. DOI: 10.1142/S0218127401002547.
  25. Kuznetsov AP, Kuznetsov SP, Ryskin NM. Nonlinear Oscillations. Moscow: Fizmatlit; 2002. 292 p. (in Russian).
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