Computation of non-smooth local centre manifolds

IMA Journal of Numerical Analysis

Volumn 25, Issue 4, 2005, Pages 698-725

  Jolly, M S ^a   Rosa, R ^b  

^a INDIANA UNIVERSITY   (United States)

^b FEDERAL UNIVERSITY OF RIO DE JANEIRO   (Brazil)

Author keywords

Centre manifolds; Computation of invariant manifolds

Indexed keywords

COMPUTATIONAL COMPLEXITY; EXPANSION; POROUS MATERIALS; TAYLOR SERIES;

ERROR ESTIMATES; HIGHER-DIMENSIONAL; INERTIAL MANIFOLDS; INVARIANT MANIFOLDS; POROUS MEDIUM; SMOOTH MANIFOLDS; TAYLOR EXPANSIONS; TEST PROBLEM;

ITERATIVE METHODS;

EID: 26444443493     PISSN: 02724979     EISSN: 14643642     Source Type: Journal    
DOI: 10.1093/imanum/dri013     Document Type: Article

References (28)

1. ARONSON, D. G. & GRAVELEAU, J. (1993) A self-similar solution to the focusing problem for the porous medium equation. Eur. J. Appl. Math., 4, 65-81.
2. CARR, J. (1981) Applications of centre manifold theory. Applied Mathematical Sciences, vol. 35. New York: Springer.
3. CASTAÑEDA, N. & ROSA, R. (1996) Optimal estimates for the uncoupling of differential equations. J. Dynam. Differ. Eqns., 8, 103-139.
4. CHOW, S.-N. & LU, K. (1988) Invariant manifolds for flows in Banach spaces. J. Differ. Eqns., 74, 285-317.
5. DELLNITZ, M. & HOHMANN, A. (1997) A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math., 75, 293-317.
6. FOIAŞ, C., SELL, G. R. & TEMAM, R. (1988) Inertial manifolds for nonlinear evolutionary equations. J. Differ. Eqns., 73, 309-353.
7. GLENDINNING, P. (1988) Global bifurcations in flows. In Bedford, T. & Swift, J. (eds), New Directions in Dynamical Systems. Cambridge: Cambridge University Press, pp. 120-149.
8. GUCKENHEIMER, J. & HOLMES, P. (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer.
9. GUCKENHEIMER, J. & VLADIMIRSKY, A. (2004) A fast method for approximating invariant manifolds. SIAM J. Appl. Dyn. Syst., 3, 232-260.
10. GUCKENHEIMER, J. & WORFOLK, P. (1993) Dynamical systems: some computational problems. Bifurcations and Periodic Orbits of Vector Fields (Montreal, PQ, 1992), vol. 408. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences. Dordrecht: Kluwer Academic Publishers, pp. 241-277.
11. HALE, J. K. (1988) Asymptotic behavior of dissipative systems. Mathematical Surveys and Monographs, vol. 25. Providence, RI: American Mathematical Society.
12. HALE, J. K. & PERELLÓ, C. (1964) The neighborhood of a singular point of functional differential equations. Contrib. Differ. Eqns., 3, 351-375.
13. HENDERSON, M. (2004) Computing invariant manifolds by integrating fat trajectories. SIAM J. Appl. Dyn. Sys. (to appear). Preprint available at http://www.research.ibm.com/people/h/henderson/.
14. HOMBURG, A. J., OSINGA, H. M. & VEGTER, G. (1995) On the computation of invariant manifolds of fixed points. Z. Angew. Math. Phys., 46, 171-187.
15. 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, 125-140.
16. [Dynamical numerical analysis (Atlanta, GA, 1995)].
17. JOHNSON, M. E., JOLLY, M. S. & KEVREKIDIS, I. G. (2001) The Oseberg transition: visualization of global bifurcations for the Kuramoto-Sivashinsky equation. Int. J. Bifur. Chaos Appl. Sci. Engng., 11, 1-18.
18. JOLLY, M. S., ROSA, R. & TEMAM, R. (2000) Accurate computations on inertial manifolds. SIAM J. Sci. Comput., 22, 2216-2238 (electronic).
19. KEVREKIDIS, I. G. (1987) A numerical study of global bifurcation in chemical dynamics. AICHE J., 33, 1850-1864.
20. KRAUSKOPF, B. & OSINGA, H. M. (1999) Two-dimensional global manifolds of vector fields. Chaos, 9, 768-774.
21. KRAUSKOPF, B. & OSINGA, H. M. (2003) Computing geodesic level sets on global (un)stable manifolds of vector fields. SIAM J. Appl. Dyn. Syst., 2, 546-569 (electronic).
22. MA, F. & KÜPPER, T. (1994) A numerical method to calculate center manifolds of ODE's. Appl. Anal., 54, 1-15.
23. MOORE, G. & HUBERT, E. (1999) Algorithms for constructing stable manifolds of stationary solutions. IMA J. Numer. Anal., 19, 375-424.
24. PERRON, O. (1929) Uber Stabilität und asmptotisches verhalten der integrale von differentialgleichungssystemen. Math. Zeit., 29, 129-160.
25. ROSA, R. (1995) Approximate inertial manifolds of exponential order. Discrete Contin. Dynam. Sys., 1, 421-448.
26. ROSA, R. & TEMAM, R. (1996) Inertial manifolds and normal hyperbolicity. ACTA Appl. Math., 45, 1-50.
27. STUART, A. M. (1994) Numerical analysis of dynamical systems. Acta Numerica, 1994, Acta Numer. Cambridge: Cambridge University Press, pp. 467-572.
28. TEMAM, R. (1997) Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68, 2nd edn. New York: Springer.