Analysis of the behavior of a random nonlinear delay discrete equation

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 6, Issue 9, 1996, Pages 1683-1690

  Cabrera, Juan L ^a   De La Rubia, F Javier ^a  

^a UNIVERSIDAD NACIONAL DE EDUCACIÓN A DISTANCIA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030237620     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S021812749600103X     Document Type: Article

References (26)

1. Aronson, D. G., Chory, M. A., Hall, G. R. & McGehee, R. P. [1982] "Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study," Commun. Math. Phys. 83, 303-354
2. Beretta, E., Bischi, G. I. & Solimano, F. [1990] "Stability in chemostat equations with delayed nutrient recycling," J. Math. Biol. 28, 99-111.
3. Beuter, A., Bélair, J., Labrie, C. & Bélair, J. [1993] "Feedback and delays in neurological diseases: A modeling study using dynamical systems," Bull. Math. Biol. 55, 525-541.
4. Cabrera, J. L. & de la Rubia, F. J. [1995] "Numerical analysis of transient behavior in the discrete random logistic equation with delay," Phys. Lett. A197, 19-24.
5. Epstein, I. R. & Luo, Y. [1991] "Differential delay equations in chemical kinetics. Nonlinear models: The cross-shaped phase diagrams and the Oregonator," J. Chem. Phys. 95, 244-254.
6. Fox, R. F., Gatland, I. R., Roy, R. & Vemury, G. [1988] "Fast accurate algorithm for numerical simulation of exponentially correlated colored noise," Phys. Rev. A38, 5938-5940.
7. Hopf, F. A., Kaplan, D. L., Gibbs, H. M. & Schoemaker, R. L. [1982] "Bifurcations to chaos in optical bistability," Phys. Rev. A25, 2172-2182.
8. Hutchinson, G. E. [1948] "Circular causal systems in ecology," Ann. N.Y. Acad. Sci. 50, 221-246.
9. Ikeda, K. & Matsumoto, K. [1987] "High dimensional chaotic behavior in systems with time delayed feedback," Physica D29, 223-235.
10. Kolmanosvskii, V. & Myshkis, A. [1992] Applied Theory of Functional Differential Equations (Kluwer Academic Publishers, Dordrecht).
11. Ladas, G. & Qian, C. [1992] "Linearized oscillations for nonautonomous delay difference equations," Cont. Math. 129, 115-125.
12. Levine, S. H., Scudo, F. M. & Plunkett, D. J. [1977] "Persistence and convergence of ecosystems: An analysis of some second order difference equations," J. Math. Biol. 4, 171-182.
13. Lin, J. & Kahn, P. B. [1980] "Random effects in population models with hereditary effects," J. Math. Biol. 10, 101.
14. Longtin, A., Milton, J. G., Bos, J. E. & Mackey, M. C. [1990] "Noise and critical behavior of the pupil light reflex at oscillation onset," Phys. Rev. A41, 6992-7005.
15. Longtin, A. [1991] "Noise-induced transitions at a Hopf bifurcation in a first-order delay-differential equation," Phys. Rev. A44, 4801-4813.
16. Mackey, M. C. & Milton, J. G. [1987] "Dynamical diseases," Ann. N. Y. Acad. Sci. 504, 16-32.
17. Mackey, M. C. & Nechaeva, I. G. [1994] "Noise and stability in differential delay equations," J. Dynamics and Differential Equations 6, 395-412.
18. Marcus, C. M. & Westervelt, R. M. [1989] "Stability of analog neural networks with delay," Phys. Rev. A39, 347-359.
19. May, R. M. [1974] "Biological populations with nonoverlapping generations: Stable points, stable cycles and chaos," Science 186, 645-647.
20. May, R. M. [1980] Lectures on Mathematics in the Life Sciences, Vol. 13 (American Mathematical Society, Providence, RI), pp. 1-64.
21. Maynard Smith, J. [1971] Mathematical Ideas in Biology (Cambridge University Press, Oxford).
22. Morimoto, Y. [1983] "Bifurcation diagram of recurrence equation of mutually coupled nervous network having refractory period," Trans. IEICE J66-A(8), 777-778.
23. Morimoto, Y. [1984] "Bifurcation diagram of recurrence equation X(t + 1) = AX(t)[1 - X(t) - X(t - 1) -X(t - 2)]," J. Phys. Soc. Jpn 53, 2460-2463.
24. Morimoto, Y. [1988] "Hopf bifurcation in the simple nonlinear recurrence equation X(t + 1) = AX(t)[1 -X(t - 1)]," Phys. Lett. A134, 179-182.
25. Morimoto, Y. [1989] "Variation of bifurcation diagram in difference equation X(t + 1) = AX(t)(1 - X(t) -BX(t - 1))," Trans. IEICE E72(1), 1-3.
26. Schell, M. & Ross, J. [1986] "Effects of time delay in rate processes," J. Chem. Phys. 85, 6489-6503.