Laguerre approximation of stable manifolds with application to connecting orbits

Mathematics of Computation

Volumn 73, Issue 245, 2004, Pages 211-242

  Moore, Gerald ^a  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Homoclinic orbits; Invariant manifolds; Laguerre polynomials; Spectral methods

Indexed keywords

EID: 0942268029     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-03-01535-7     Document Type: Article

References (24)

1. H. Amann, Ordinary Differential Equations, Studies in Mathematics 13, de Gruyter, 1990. MR 91e:34001
2. Z. Bashir-Ali, J.R. Cash and G. Moore, Numerical experiments with algorithms for connecting orbits, in preparation.
3. W-J. Beyn, The numerical computation of connecting orbits in dynamical systems., I.M.A. Jour. Num. Anal. 9 (1990), 379-405. MR 011:65146
4. W-J. Beyn, Numerical methods for dynamical systems, Advances In Numerical Analysis (W. Light, ed.), Clarendon Press, Oxford, 1991, pp. 175-236. MR 92f:65005
5. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Mechanics, Springer, New York, 1987. MR 89m:76004
6. A.R. Champneys, Y.A. Kuznetsov and B. Sandstede, A numerical toolbox for homoclinic bifurcation analysis, Int. J. Bif. & Chaos 6 (1996), 867-887. MR 98b:65094
7. P.J. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, 1975. MR 56:7119
8. E.J. Doedel, A.R. Champneys, T.F. Fairgrieve, Y.A. Kuznetsov, B. Sandstede & X.J. Wang, AUTO 97: Continuation And Bifurcation Software for Ordinary Differential Equations (with HomCont), Technical Report, California Institute Of Technology, 1998.
9. M.J. Friedman and E.J. Doedel, Numerical computation and continuation of invariant manifolds connecting fixed points, SIAM J. Numer. Anal. 28 (1991), 789-808. MR 98e:34058
10. D. Funaro, Computational aspects of pseudospectral Laguerre approximations, Applied Numerical Mathematics 6 (1989/90), 447-457. MR 91m:65074
11. W. Gautschi, Orthogonal polynomials: applications and computation, Acta Numerica 5 (A. Iserles, ed.), Cambridge University Press, 1996, pp. 45-119. MR 99j:65019
12. G.H. Golub and C.F. van Loan, Matrix Computations, John Hopkins, Baltimore, 1989. MR 90d:65055
13. G.H. Golub and J.H. Welsch, Calculation of Gauss quadrature rules, Math. Comp. 23 (1969), 221-230. MR 39:6513
14. J. Guckenheimer and P.H. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences Vol. 42, Springer, 1983. MR 85f:58002
15. M.W. Hirsch and S. Smale, Differential Equations. Dynamical Systems and Linear Algebra, Academic Press, San Diego, 1974. MR 58:6484
16. Y. Liu, L. Liu and T. Tang, The numerical computation of connecting orbits in dynamical systems: a rational spectral approach, J. Comput. Phys. 111 (1994), 373-380. MR 95a:65109
17. Y. Maday, B. Pernaud-Thomas and H. Vandeven, Reappraisal of Laguerre type spectral methods, Rech. Aerosp. (1985-86), 13-35. MR 88b:65135
18. C. Mavriplis, Laguerre polynomials for infinite-domain spectral elements, J. Comput. Phys. 80 (1989), 480-488. MR 90f:65228
19. G. Moore, Computation and parametrisation of periodic and connecting orbits., I.M.A. Jour. Num. Anal. 15 (1995), 245-263. MR 96a:34087
20. G. Moore, Geometric methods for computing invariant manifolds, Applied Numerical Mathematics 17 (1995), 319-331. MR 96g:65082
21. G. Moore and E. Hubert, Algorithms for constructing stable manifolds of stationary solutions, I.M.A. Jour. Num. Anal. 19 (1999), 375-424. MR 2000f:65066
22. n f(x)dx, Math. Tables Aids Comp. 13 (1959), 285-294. MR 21:6713
23. D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory, Academic Press, San Diego, 1989. MR 90f:58048
24. M. Shub, Global Stability of Dynamical Systems, Springer, New York, 1987. MR 87m:58086