Boundary crises arising from saddle-node bifurcations in periodically forced models

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 6, Issue 4, 1996, Pages 647-659

  Foster, E A D ^a  

^a MEMORIAL UNIVERSITY OF NEWFOUNDLAND   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0039928460     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127496000321     Document Type: Article

References (45)

1. Bardi, M. [1981] "Predator-prey models in periodically fluctuating environments," J. Math. Biol. 12, 127-140.
2. Berryman, A. A. [1991] "Can economic forces cause ecological chaos? The case of the northern California Dungeness crab fishery," Oikos 62, 106-109.
3. Blom, J. G., de Bruin, R., Grasman, J. & Verwer, J. G. [1981] "Forced prey-predator oscillations," J. Math. Biol. 12, 141-152.
4. Braun, T. & Lisbôa, J. A. [1994] "Characterization of homoclinic chaos in a glow discharge through return maps," Int. J. Bifurcation and Chaos 4, 1483-1493.
5. Brown, R. & Chua, L. [1991] "Horseshoes in the twist- and-flip map," Int. J. Bifurcation and Chaos 1, 235-252.
6. Cushing, J. M. [1977] "Periodic time-dependent predator-prey systems," SIAM J. Appl. Math. 32, 82-95.
7. Devaney, R. L. [1989] An Introduction to Chaotic Dynamical Systems, Second Edition (Addison-Wesley Reading, MA).
8. Dawson, S. P., Grebogi, C., Yorke, J. A., Kan, I. & Koçak, H. [1992] "Antimonotonicity: Inevitable reversals of period doubling cascades," Phys. Lett. A162, 249-254.
9. Dawson, S. P., Grebogi, C. & Koçak, H. [1993] "Geometric mechanism for antimonotonicity in scalar maps with two critical points," Phys. Rev. E48, 1676-1682.
10. Fan, Y. S. & Chay, T. R. [1993] "Crisis transitions in excitable cell models," Chaos Solitons Fractals 3, 603-615.
11. Fan, Y. S. & Chay, T. R. [1995] "Crisis and topological-entropy," Phys. Rev. E51, 1012-1019.
12. Fan, Y. S. & Holden, A. V. [1993] "Bifurcations, burstings, chaos and crises in the Rose-Hindmarsh model for neuronal activity," Chaos Solitons Fractals 3, 439-449.
13. Grebogi, C., Ott, E. & Yorke, J. A. [1982] "Chaotic attractors in crisis," Phys. Rev. Lett. 48, 1507-1513.
14. Grebogi, C., Ott, E. & Yorke, J. A. [1983] "Crises, sudden changes in chaotic attractors and chaotic transients," Physica D7, 181-200.
15. Grebogi, C., Ott, E. & Yorke, J. A. [1986] "Critical exponents of chaotic transients in nonlinear dynamical systems," Phys. Rev. Lett. 57, 1284-1288.
16. Grebogi, C., Ott, E., Romeiras, F. & Yorke, J. A. [1987] "Critical exponents for crisis-induced intermittency," Phys. Rev. A36, 5365-5380.
17. Griffiths, D. F., Sweby, P. K. & Yee, H. C. [1992] "On spurious asymptotic numerical solutions of explicit Runge-Kutta methods," IMA J. Numer. Anal. 12, 319-338.
18. Higgins, J. [1964] "A chemical mechanism for oscillation glycolytic intermediates in yeast cells," Proc. Natl. Acad. Sci. USA 51, 989-994.
19. Holden, A. V. & Fan, Y. S. [1992a] "From simple to simple bursting oscillatory behaviour via chaos in the Rose-Hindmarsh model for neuronal activity," Chaos Solitons Fractals 2, 221-236.
20. Holden, A. V. & Fan, Y. S. [1992b] "From simple to complex oscillatory behaviour via intermittent chaos in the Rose-Hindmarsh model for neuronal activity," Chaos Solitons Fractals 2, 349-369.
21. Holden, A. V. & Fan, Y. S. [1992c] "Crisis-induced chaos in the Rose-Hindmarsh model for neuronal activity," Chaos Solitons Fractals 2, 583-595.
22. Hsu, G. H., Ott, E. & Grebogi, C. [1988] "Strange saddles and the dimensions of their invariant manifolds," Phys. Lett. A127, 199-204.
23. Humphries, A. R. [1993] "Spurious solutions of numerical methods for initial value problems," IMA J. Numer. Anal. 13, 263-290.
24. Inoue, M. & Kamifukumoto, H. [1984] "Scenarios leading to chaos in a forced Lotka-Volterra model," Prog. Theor. Phys. 71, 930-937.
25. Kuznetsov, Yu. A., Muratori, S. & Rinaldi, S. [1992] "Bifurcations and chaos in a periodic predator-prey model," Int. J. Bifurcation and Chaos 2, 117-128.
26. Lai, Y. C., Grebogi, C. & Yorke, J. A. [1992] "Sudden change in the size of chaotic attractors: How does it occur?," in Applied Chaos, eds. Kim, J. H. & Stringer, J. (Wiley, New York).
27. Mottoni, P. de & Schiaffino, A. [1981] "Competition systems with periodic coefficients: a geometric approach," J. Math. Biol. 11, 319-335.
28. Myerscough, M. R., Gray, B. F., Hogarth, W. L. & Norbury, J. [1992] "An analysis of an ordinary differential equation model for a two-species predator-prey system with harvesting and stocking," J. Math. Biol. 30, 389-411.
29. Parker, T. S. & Chua, L. O. [1989] Practical Numerical Algorithms for Chaotic Systems (Springer-Verlag, New York).
30. Pavlou, S. & Kevrekidis, I. G. [1992] "Microbial predation in a periodically operated chemostat: A global study of the interaction between natural and externally imposed frequencies," Math. Biosci. 108, 1-55.
31. Rinaldi, S., Muratori, S. & Kuznetsov, Yu. A. [1993] "Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities," Bull. Math. Biol. 55, 15-36.
32. Rosenzweig, M. & MacArthur, R. H. [1963] "Graphical representation and stability conditions of predator-prey interaction," Am. Nat. 98, 209-223.
33. Rosenzweig, M. [1971] "Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time," Science 171, 385-387.
34. Sabin, G. C. W. & Summers, D. [1993] "Chaos in a periodically forced predator-prey ecosystem model," Math. Biosci. 113, 91-113.
35. Schaffer, W. M. & Kot, M. [1985] "Nearly one-dimensional dynamics in an epidemic," J. Math. Biol. 112, 403-427.
36. Schnabl, W., Stadler, P. F., Forst, C. & Schuster, P. [1991] "Full characterization of a strange attractor. Chaotic dynamics in low-dimensional replicator systems," Physica D48, 65-90.
37. Sel'kov, E. E. [1968] "Self-oscillations in glycolysis," Eur. J. Biochem. 4, 79-86.
38. Shaw, R. [1981] "Strange attractors, chaotic behavior, and information flow," Z. Naturforsch. A36, 80-112.
39. Simó, C. [1979] "On the Hénon-Pomeau attractor," J. Stat. Phys. 21, 465-476.
40. Simoyi, R. H., Wolf, A. & Swinney, H. L. [1982] "One-dimensional dynamics in a multicomponent chemical reaction," Phys. Rev. Lett. 49, 245-266.
41. Tomita, K. & Daido, H. [1980] "Possibility of chaotic behavior and multi-basins in forced glycolytic oscillations," Phys. Lett. A79, 133-137.
42. Tomita, K. & Kai, T. [1979] "Chaotic response of a limit cycle," J. Stat. Phys. 21, 65-85.
43. Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A. [1985] "Determining Lyapunov exponents from a time series," Physica D16, 285-317.
44. Wrzosek, D. M. [1990] "Limit cycles in predator-prey models," Math. Biosci. 98, 1-12.
45. Yorke, J. A. & Yorke, E. D. [1979] "Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model," J. Stat. Phys. 21, 263-277.