The Oseberg transition: Visualization of global bifurcations for the Kuramoto-Sivashinsky equation

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 11, Issue 1, 2001, Pages 1-18

  Johnson, M E ^a   Jolly, M S ^b   Kevrekidis, I G ^c  

^a UNIVERSITY OF MINNESOTA   (United States)

^b INDIANA UNIVERSITY   (United States)

^c PRINCETON UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034747501     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127401001979     Document Type: Review

References (36)

1. Armbruster, D., Guckenheimer, J. & Holmes, P. [1989] "Kuramoto-Sivashinsky dynamics on the center-unstable manifold," SIAM J. Appl. Math. 49(3), 676-691.
2. Brogger, A. & Shetelig, H. [1971] The Viking Ships (Twayne, NY).
3. Brown, H. S. & Kevrekidis, I. G. [1996] "Modulated traveling waves for the Kuramoto-Sivashinsky equation," in Pattern Formation: Symmetry Methods and Applications (Waterloo, ON, 1993) (Amer. Math. Soc., Providence, RI), pp. 45-66.
4. Brown, H. S., Kevrekidis, I. G. & Jolly, M. S. [1991] "A minimal model for spatio-temporal patterns in thin film flow," in Patterns and Dynamics in Reactive Media (Minneapolis, MN, 1989) (Springer, NY), pp. 11-31.
5. Chang, H.-C. [1986] "Traveling waves on fluid interfaces: Normal form analysis of the Kuramoto-Sivashinsky equation," Phys. Fluids 29(10), 3142-3147.
6. Dankowicz, H., Holmes, P., Berkooz, G. & Elezgaray, J. [1996] "Local models of spatio-temporally complex fields," Phys. D90(4), 387-407.
7. Dawson, S. & Mancho, A. [1997] "Collections of heteroclinic cycles in the Kuramoto-Sivashinsky equation," Phys. D100(3&4), 231-256.
8. Doedel, E. [1981] "AUTO: A program for the automatic bifurcation analysis of autonomous systems," Cong. Num. 30, 265-284.
9. Foiaş, C., Jolly, M., Kevrekidis, I., Sell, G. & Titi, E. [1988a] "On the computation of inertial manifolds," Phys. Lett. A131(7&8), 433-436.
10. Foiaş, C., Nicolaenko, B., Sell, G. R. & Temam, R. [1988b] ̧Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension,̧ J. Math. Pure Appl. (9) 67(3), 197-226.
11. Ghrist, R. W., Holmes, P. J. & Sullivan, M. C. [1997] Knots and Links in Three-Dimensional Flows (Springer-Verlag, Berlin).
12. Glendinning, P. & Sparrow, C. [1984] "Local and global behavior near homoclinic orbits," J. Statist. Phys. 35(5&6), 645-696.
13. Golubitsky, M. & Schaeffer, D. [1985] Singularities and Groups in Bifurcation Theory, Vol. I, Applied Mathematical Sciences, Vol. 51 (Springer-Verlag, NY).
14. Golubitsky, M., Stewart, I. & Schaeffer, D. [1988] Singularities and Groups in Bifurcation Theory, Vol. II, Applied Mathematical Sciences, Vol. 69 (Springer-Verlag, NY).
15. Greenhill, B. [1976] Archaeology of the Boat (Wesleyan University Press, Middletown, Conn).
16. Hyman, J. M. & Nicolaenko, B. [1986] "The Kuramoto-Sivashinsky equation: A bridge between PDEs and dynamical systems," Phys. D18(1-3), 113-126; Solitons and Coherent Structures (Santa Barbara, Calif., 1985).
17. Johnson, M. [1998] "Computer-assisted studies and visualization of nonlinear phenomena: Two-dimensional invariant manifolds, global bifurcations & robustness of global attractors," Ph.D. dissertation, Princeton University, Program in Applied and Computational Mathematics.
18. Johnson, M. E., Jolly, M. S. & Kevrekidis, I. G. [1997] "Two-dimensional invariant manifolds and global bifurcations: Some approximation and visualization studies," Numer. Algorithms 14(1-3); 125-140; Dynamical Numerical Analysis (Atlanta, GA, 1995).
19. Jolly, M., Kevrekidis, I. & Titi, E. [1990] "Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: Analysis and computations," Phys. D44(1&2), 38-60.
20. Jolly, M., Rosa, R. & Temam, R. [2000] "Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky equation," Adv. Diff. Eqs. 5, 31-66.
21. Jolly, M., Rosa, R. & Temam, R. [2001] "Accurate computations on inertial manifolds," SIAM J. Scient. Comp., to appear.
22. Kent, P. & Elgin, J. [1992] "Travelling-waves of the Kuramoto-Sivashinsky equation: Period-multiplying bifurcations," Nonlinearity 5(4), 899-919.
23. Kevrekidis, I., Nicolaenko, B. & Scovel, J. [1990] "Back in the saddle again: A computer assisted study of the Kuramoto-Sivashinsky equation," SIAM J. Appl. Math. 50(3), 760-790.
24. Krauskopf, B. & Osinga, H. [1999] "Two-dimensional global manifolds of vector fields," Chaos 9(3), 768-774.
25. Kuramoto, Y. & Tsuzuki, T. [1976] "Persistent propagation of concentration waves in dissipative media far from thermal equilibrium," Prog. Theo. Phys. 55(2), 356-369.
26. Liu, X. [1991] "Gevrey class regularity and approximate inertial manifolds for the Kuramoto-Sivashinsky equation," Phys. D50(1), 135-151.
27. Lorenz, E. [1963] "Deterministic nonperiodic flows," J. Atmospheric Sci. 20, 130-141.
28. Palis Jr., J. & de Melo, W. [1982] Geometric Theory of Dynamical Systems (Springer-Verlag, NY-Berlin).
29. Robinson, J. [1994] "Inertial manifolds for the Kuramoto-Sivashinsky equation," Phys. Lett. A184(2), 190-193.
30. Rosa, R. [1995] "Approximate inertial manifolds of exponential order," Discrete Contin. Dyn. Syst. 1(3), 421-448.
31. Scovel, J., Kevrekidis, I. G. & Nicolaenko, B. [1988] "Scaling laws and the prediction of bifurcations in systems modeling pattern formation," Phys. Lett. A130, 73-80.
32. Sivashinsky, G. I. [1980] "On flame propagation under conditions of stoichiometry," SIAM J. Appl. Math. 39(1), 67-82.
33. Swift, J. & Weisenfeld, K. [1984] "Supression of period doubling in symmetric systems," Phys. Rev. Lett. 52(9), 705-708.
34. Temam, R. [1988] Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Vol. 68 (Springer-Verlag, NY).
35. Temam, R. & Wang, X. [1994] "Estimates on the lowest dimension of inertial manifolds for the Kuramoto-Sivashinsky equation in the general case," Diff. Int. Eqs. 7(3&4), 1095-1108.
36. Wittenberg, R. W. & Holmes, P. [1999] "Scale and space localization in the Kuramoto-Sivashinsky equation," Chaos 9(2), 452-465.