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Bifurcations and chaos in a predator-prey model with delay and a laser-diode system with self-sustained pulsations. (English) Zbl 1033.37048
Summary: Hopf bifurcations in two models, a predator-prey model with delay terms modeled by “weak generic kernel $a \exp(-at)$” and a laser diode system, are considered. The periodic orbit immediately following the Hopf bifurcation is constructed for each system using the method of multiple scales, and its stability is analyzed. Numerical solutions reveal the existence of stable periodic attractors, attractors at infinity, as well as bounded chaotic dynamics in various cases. The dynamics exhibited by the two systems is contrasted and explained on the basis of the bifurcations occurring in each.

37N25Dynamical systems in biology
37N20Dynamical systems in other branches of physics
37G15Bifurcations of limit cycles and periodic orbits
37D45Strange attractors, chaotic dynamics
34K18Bifurcation theory of functional differential equations
78A60Lasers, masers, optical bistability, nonlinear optics
92D25Population dynamics (general)
Full Text: DOI
[1] Cushing, J. M.: Integrodifferential equations and delay models in population dynamics. Lecture notes in biomathematics 20 (1977) · Zbl 0363.92014
[2] Smitalova, K.; Sujan, S.: A mathematical treatment of dynamical models in biological science. (1991)
[3] Macdonald, N.: Time lags in biological models. Lecture notes in biomathematics 27 (1978) · Zbl 0403.92020
[4] Farkas, M.: Stable oscillations in a predator--prey model with time lag. J. math. Anal. appl. 102, 175-188 (1984) · Zbl 0536.92023
[5] Davis, H. T.: Introduction to nonlinear differential and integral equations. (1962) · Zbl 0106.28904
[6] El-Owaidy, H.; Ammar, A. A.: Stable oscillations in a predator--prey model with time lag. J. math. Anal. appl. 130, 191-199 (1988) · Zbl 0631.92015
[7] Vogel, T.: Systemes evolutifs. (1965) · Zbl 0161.05802
[8] Macdonald, N.: II. bifurcation theory. Biosciences 33, 227-234 (1977) · Zbl 0354.92036
[9] Marsden, J. E.; Mccracken, M.: The Hopf bifurcation and its applications. (1976) · Zbl 0346.58007
[10] Murray, J. D.: Mathematical biology. (1989) · Zbl 0682.92001
[11] Roos E. Predator--prey models with distributed delay. MS thesis, University of Central Florida, Orlando, 1991
[12] Choudhury, S. R.: On bifurcations and chaos in predator--prey models with delay. Chaos, solitons and fractals 2, 393-409 (1992) · Zbl 0753.92022
[13] Ueno, M.; Lang, R.: Conditions for self-sustained pulsation and bistability in semiconductor lasers. 58, 1689-1692 (1985)
[14] Wang, X.; Li, G.; Ih, C. H.: Microwave/millimeter-wave frequency subcarrier lightwave modulations based on self-sustained pulsation of a laser diode. J. lightwave tech. 11, 309-315 (1993)
[15] Nayfeh, A. H.; Balachandran, B.: Applied nonlinear dynamics. (1995) · Zbl 0848.34001
[16] Seydel, R.: From equilibrium to chaos. (1988) · Zbl 0652.34059
[17] Abarbanel, H. D. I.; Rabinovich, M. I.; Sushchik, M. M.: Introduction to nonlinear dynamics for physicists. (1993)
[18] Kincaid, D.; Cheney, W.: Numerical analysis. (1991) · Zbl 0745.65001
[19] Nayfeh, A. H.; Mook, D. T.: Nonlinear oscillations. (1979) · Zbl 0418.70001