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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.

MSC:

37N25 Dynamical systems in biology
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34K18 Bifurcation theory of functional-differential equations
78A60 Lasers, masers, optical bistability, nonlinear optics
92D25 Population dynamics (general)
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[1] Cushing, J. M., Integrodifferential equations and delay models in population dynamics, (Lecture Notes in Biomathematics, vol. 20 (1977), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0363.92014
[2] Smitalova, K.; Sujan, S., A mathematical treatment of dynamical models in biological science (1991), Ellis Horwood: Ellis Horwood New York, and references therein · Zbl 0751.92011
[3] MacDonald, N., Time lags in biological models, (Lecture Notes In Biomathematics, vol. 27 (1978), Springer-Verlag: Springer-Verlag Berlin) · 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), Dover: Dover New York
[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), Gautier-Villars: Gautier-Villars Paris · Zbl 0132.21701
[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), Springer-Verlag: Springer-Verlag New York · Zbl 0346.58007
[10] Murray, J. D., Mathematical biology (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0682.92001
[11] Roos E. Predator-prey models with distributed delay. MS thesis, University of Central Florida, Orlando, 1991; 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), Wiley: Wiley New York
[16] Seydel, R., From equilibrium to chaos (1988), Elsevier: Elsevier New York · Zbl 0652.34059
[17] Abarbanel, H. D.I.; Rabinovich, M. I.; Sushchik, M. M., Introduction to nonlinear dynamics for physicists (1993), World Scientific: World Scientific Singapore
[18] Kincaid, D.; Cheney, W., Numerical analysis (1991), Brooks/Cole: Brooks/Cole Pacific Grove, CA
[19] Nayfeh, A. H.; Mook, D. T., Nonlinear oscillations (1979), Wiley: Wiley New York
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