Features of Bifurcations of Periodic Solutions of the Ikeda Equation

    accepted 24 August 2018

    2018, Vol. 14, no. 3, pp.  301-324

    Author(s): Kubyshkin E. P., Moriakova A. R.

    We study equilibrium states and bifurcations of periodic solutions from the equilibrium state of the Ikeda delay-differential equation well known in nonlinear optics. This equation was proposed as a mathematical model of a passive optical resonator in a nonlinear environment. The equation, written in a characteristic time scale, contains a small parameter at the derivative, which makes it singular. It is shown that the behavior of solutions of the equation with initial conditions from the fixed neighborhood of the equilibrium state in the phase space of the equation is described by a countable system of nonlinear ordinary differential equations. This system of equations has a minimal structure and is called the normal form of the equation in the neighborhood of the equilibrium state. The system of equations allows us to pick out one “fast” and a countable number of “slow” variables and apply the averaging method to the system obtained. It is shown that the equilibrium states of a system of averaged equations with “slow” variables correspond to periodic solutions of the same type of stability. The possibility of simultaneous bifurcation of a large number of stable periodic solutions(multistability bifurcation) is shown. With further increase in the bifurcation parameter each of the periodic solutions becomes a chaotic attractor through a series of period-doubling bifurcations. Thus, the behavior of the solutions of the Ikeda equation is characterized by chaotic multistability.
    Keywords: Ikeda equation, periodic solutions, bifurcation of multistability, chaotic multistability
    Citation: Kubyshkin E. P., Moriakova A. R., Features of Bifurcations of Periodic Solutions of the Ikeda Equation, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 3, pp.  301-324

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    [1] Ikeda, K., “Multiple-Valued Stationary State and Its Instability of the Transmitted Light a Ring Cavity System”, Opt. Commun., 30:2 (1979), 257–261  crossref  adsnasa
    [2] Ikeda, K., Daido, H., and Akimoto, O., “Optical Turbulence: Chaotic Behavior of Transmitted Light from a Ring Cavity”, Phys. Rev. Lett., 45:9 (1980), 709–712  crossref  adsnasa
    [3] Ikeda, K. and Matsumoto, K., “High-Dimensional Chaotic Behavior in System with Time-Delayed Feedback”, Phys. D, 29:1–2 (1987), 223–235  crossref  mathscinet  zmath
    [4] Sprott, J. C., Elegant Chaos: Algebraically Simple Chaotic Flows, World Sci., Singapore, 2010, 304 pp.  mathscinet  zmath  adsnasa
    [5] Ponomarenko, V. I. and Prokhorov, M. D., “Recovering Parameters of the Ikeda Equation from Noisy Time Series”, Tech. Phys. Lett., 31:3 (2005), 252–254  crossref  mathscinet  adsnasa; Pis'ma Zh. Tekh. Fiz., 31:6 (2005), 73–78 (Russian)  mathscinet
    [6] Larger, L., Goedgebuer, J.-P., and Udaltsov, V., “Ikeda-Based Nonlinear Delayed Dynamics for Application to Secure Optical Transmission Systems Using Chaos”, C. R. Math. Acad. Sci. Paris, 5:6 (2004), 669–681
    [7] Kubyshkin, E. P. and Nazarov, A. Yu., “Analysis of Oscillatory Solutions of a Nonlinear Singularly Perturbed Differential-Difference Equation”, Vestn. Nizhegorodsk. Univ., 5:2 (2010), 118–125 (Russian)
    [8] Kubyshkin, E. P. and Moriakova, A. R., “Analysis of Bifurcations of Periodic Solutions of Ikeda Equation”, Nonlinear Phenomena in Complex Systems, 20:1 (2017), 40–49  mathscinet  zmath
    [9] Bellman, R. and Cooke, K. L., Differential-Difference Equations, Acad. Press, New York, 1963, 462 pp.  mathscinet  zmath
    [10] Krein, S. G, Linear Differential Equations in Banach Space, AMS, Providence, R.I., 1972  mathscinet
    [11] Shimanov, S. N., “On the Vibration Theory of Quasilinear Systems with Lag”, J. Appl. Math. Mech., 23:5 (1959), 1198–1208  crossref  mathscinet  zmath; Prikl. Mat. Mekh., 23:5 (1959), 836–844 (Russian)  zmath
    [12] Krasnosel'skii, M. A., Vainikko, G. M., Zabreyko, P. P., Ruticki, Ya. B., and Stetsenko, V. Ya., Approximate Solution of Operator Equation, Wolters-Noordhoff, Groningen, 1972, 496 pp.  mathscinet
    [13] Neimark, Yu. I., “D-Partitions of the Space of Quasipolynomials (to the Stability of Linearized Distributed Systems)”, Prikl. Mat. Mekh., 13:4 (1949), 349–380 (Russian)  mathscinet
    [14] Glyzin, D. S., The Program Package for the Analysis of Dynamic Systems “Tracer”, Appl. no. 2008610548 dated Feb 14, 2008, Certificate of State Registration of the Computer Program no. 2008611464, Registered in the Register of Computer Programs 24.03.2008 (Russian)

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