Dynamics of a System of Two Simplest Oscillators with Compactly Supported Nonlinear Feedbacks
In this paper, we consider a singularly perturbed system of two differential equations with delay, simulating two coupled oscillators with a nonlinear feedback. The feedback function is assumed to be compactly supported and piecewise-continuous and it is assumed that its sign is constant. In this paper, we prove the existence of relaxation periodic solutions and make conclusions about their stability. Using a special large-parameter method, we construct asymptotics of all solutions of the considered system under the assumption that the initial-value conditions belong to a certain class. Using this asymptotics, we construct a special mapping principally describing the dynamics of the original model. It is shown that the dynamics changes fundamentally as the coupling coefficient decreases: we have a stable homogeneous periodic solution if the coupling coefficient is on the order of unity and the dynamics become more complex as the coupling coefficient decreases (it is described by a special map). For small values of the coupling, we show that there are values of the parameters such that several different stable relaxation periodic regimes coexist in the original problem.
Keywordsasymptotics stability large parameter relaxation oscillations periodic solution
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- 1.Dmitriev, A.S. and Kislov, V.Ya., Stokhasticheskie kolebaniya v radiotekhnike (Stochastic Oscillations in Radio Engineering), Moscow: Nauka, 1989.Google Scholar
- 2.Rozhdestvenskiy, V.V. and Struchkov, I.N., Transient chaos in an oscillator of stochastic oscillations with rigid excitation and even nonlinearity, J. Tech. Phys., 1992, vol. 62, no. 10, pp. 102–110.Google Scholar
- 3.Struchkov, I.N., Transient chaos in an aperiodic oscillator with delay, Radiotekh. Elektron., 1993, vol. 38, no. 3, pp. 454–463.Google Scholar
- 4.Dmitriev, A.S. and Kaschenko, S.A., Dynamics of a generator with delayed feedback and a low-order filter of the second order, Radiotekh. Elektron., 1989, vol. 34, no. 12, pp. 24–39.Google Scholar
- 5.Dmitriev, A.S. and Kaschenko, S.A., Asymptotics of irregular oscillations in a self-sustained oscillator model with delayed feedback, Dokl. Math., 1993, vol. 328, pp. 157–160.Google Scholar
- 8.Sharkovsky, A.N., Maistrenko, Yu.L., and Romanenko, E.Yu., Difference Equations and Their Applications, Springer Science & Business Media, 2012Google Scholar
- 9.Kaschenko, S.A., Relaxation oscillations in a system with delays modeling the predator-prey problem, Model. Anal. Inf. Syst., 2013, vol. 21, no. 3, pp. 52–98.Google Scholar