Dynamics of a singularly perturbed system of two differential equations with delay

I. S. Kashchenko   ; E. V. Krivets

Teoretičeskaâ i matematičeskaâ fizika, Tome 207 (2021) no. 3, pp. 424-437

Résumé

We study a two-dimensional singularly perturbed system with delay, which is a simplification of models used in laser physics. We analyze several cases of a small parameter multiplying the derivative in the first equation and investigate the behavior of solutions in a neighborhood of a stationary point when the system parameters pass through bifurcation values. Methods for local asymptotic analysis are used to construct special nonlinear equations describing the structure of solutions and the asymptotic approximation of solutions of the original problem.

Détail

Keywords: dynamics, asymptotics, normal form, delay.
Mots-clés : singular perturbation

Bibliographie

[1] M. C. Mackey, L. Glass, “Oscillation and chaos in physiological control systems”, Science, 197:4300 (1977), 287–289 | DOI

[2] R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties”, IEEE J. Quantum Electron., 16:3 (1980), 347–355 | DOI

[3] H. Haken, Brain Dinamics. Synchronization and Activity Patterns in Pulse-Coupled Neural Nets with Delays and Noise, Springer Series in Synergetics, Springer, Berlin, 2002 | DOI | MR

[4] T. Erneux, Applied Delay Differential Equations, Surveys and Tutorials in the Applied Mathematical Sciences, 3, Springer, New York, 2009 | DOI | MR

[5] S. A. Kaschenko, V. V. Maiorov, Modeli volnovoi pamyati, Librokom, M., 2009 | DOI | MR

[6] V. Kolmanovskii, A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer, Berlin, 2013 | MR

[7] A. S. Karavaev, Yu. M. Ishbulatov, V. I. Ponomarenko, B. P. Bezruchko, A. R. Kiselev, M. D. Prokhorov, “Autonomic control is a source of dynamical chaos in the cardiovascular system”, Chaos, 29:12 (2019), 121101, 8 pp. | DOI | MR

[8] J. K. Hale, S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer, New York, 1993 | DOI | MR

[9] E. V. Grigorieva, H. Haken, S. A. Kaschenko, “Theory of quasiperiodicity in model of lasers with delayed optoelectronic feedback”, Opt. Commun., 165:4–6 (1999), 279–292 | DOI

[10] A. A. Kashchenko, “Multistability in a system of two coupled oscillators with delayed feedback”, J. Differ. Equ., 266:1 (2019), 562–579 | DOI | MR

[11] S. A. Kaschenko, “Dinamika logisticheskogo uravneniya, soderzhaschego zapazdyvanie”, Matem. zametki, 98:1 (2015), 85–100 | DOI | DOI | MR

[12] A. A. Kashchenko, “Relaxation modes of a system of diffusion coupled oscillators with delay”, Commun. Nonlinear Sci. Numer. Simul., 93 (2021), 105488, 10 pp. | DOI | MR

[13] V. O. Golubenets, “Relaksatsionnye kolebaniya v logisticheskom uravnenii s nepostoyannym zapazdyvaniem”, Matem. zametki, 107:6 (2020), 833–847 | DOI | DOI | MR

[14] S. Yanchuk, S. Ruschel, J. Sieber, M. Wolfrum, “Temporal dissipative solitons in time-delay feedback systems”, Phys. Rev. Lett., 123:5 (2019), 053901, 6 pp. | DOI | MR

[15] L. Larger, B. Penkovsky, Y. Maistrenko, “Virtual chimera states for delayed-feedback systems”, Phys. Rev. Lett., 111:5 (2013), 054103, 5 pp. | DOI

[16] E. V. Grigoreva, S. A. Kaschenko, “Normalizovannye kraevye zadachi v modeli optiko-elektronnogo ostsillyatora s zapazdyvaniem”, Izvestiya vuzov. PND, 28:4 (2020), 361–382 | DOI

[17] I. Kashchenko, S. Kaschenko, “Infinite process of forward and backward bifurcations in the logistic equation with two delays”, Nonlinear Phenom. Complex Syst., 22:4 (2019), 407–412 | DOI

[18] A. G. Vladimirov, D. V. Turaev, “Model for passive mode locking in semiconductor lasers”, Phys. Rev. A, 72:3 (2005), 033808, 13 pp. | DOI

[19] A. Pimenov, J. Javaloyes, S. V. Gurevich, A. G. Vladimirov, “Light bullets in a time-delay model of a wide-aperture mode-locked semiconductor laser”, Philos. Trans. Roy. Soc. A, 376:2124, 20170372, 14 pp. | DOI | MR

[20] A. G. Vladimirov, D. Rachinskii, M. Wolfrum, “Modeling of passively mode-locked semiconductor lasers”, Nonlinear Laser Dynamics: From Quantum Dots to Cryptography, ed. K. Lüdge, John Wiley and Sons, New York, 2012, 183–216 | DOI

[21] S. Slepneva, B. Kelleher, B. O'Shaughnessy, S. P. Hegarty, A. G. Vladimirov, G. Huyet, “Dynamics of Fourier domain mode-locked lasers”, Opt. Express, 21:16 (2013), 19240–19251 | DOI

[22] S. A. Kaschenko, “Primenenie metoda normalizatsii k izucheniyu dinamiki differentsialno-raznostnogo uravneniya s malym mnozhitelem pri proizvodnoi”, Differents. uravneniya, 25:8 (1989), 1448–1451 | MR | Zbl

[23] I. S. Kaschenko, “Lokalnaya dinamika uravnenii s bolshim zapazdyvaniem”, Zh. vychisl. matem. i matem. fiz., 48:12 (2008), 2141–2150 | DOI | MR

[24] A. Fridman, Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa, Mir, M., 1968