Invariant manifolds in dissipative dynamical systems

Acta Applicandae Mathematicae

Volumn 87, Issue 1-3, 2005, Pages 229-244

  Verhulst, Ferdinand ^a  

^a UTRECHT UNIVERSITY   (Netherlands)

Author keywords

Averaging; Invariant manifolds; Normalisation; Tori

Indexed keywords

ALGORITHMS; COMPUTATIONAL GEOMETRY; MATRIX ALGEBRA; PARTIAL DIFFERENTIAL EQUATIONS; SPHERES; VECTORS;

AVERAGING; INVARIANT MANIFOLDS; NORMALIZATIONS; TORI;

DYNAMICS;

EID: 21044449042     PISSN: 01678019     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10440-005-1159-4     Document Type: Conference Paper

References (26)

1. Afraimovich, V. S. and Shil'nikov, L. P. (1991) Invariant two-dimensional tori, their breakdown and stochasticity, In: Amer. Math. Soc. Transl. 149, pp. 201-212.
2. Aronson, D. G., Chory, M. A., Hall, G. R. and McGehee, R. P. (1982) Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study, Comm. Math. Phys. 83, 303-354.
3. Bakri, T., Nabergoj, R., Tondl, A. and Verhulst, F. (2004) Parametric excitation in nonlinear dynamics, Int. J. Nonlinear Mech. 39, 311-329.
4. Bogoliubov, N. N. and Mitropolsky, Yu. A. (1961) Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York.
5. Bogoliubov, N. N. and Mitropolsky, Yu. A. (1963) The method of integral manifolds in nonlinear mechanics, In: Contributions to Differential Equations, Vol. 2, Interscience-Wiley, pp. 123-196.
6. Broer, H. W., Huitema, G. B. and Sevryuk, M. B. (1996) Quasi-Periodic Motions in Families of Dynamical Systems: Order Amidst Chaos, Lecture Notes in Math. 1645, Springer-Verlag.
7. Broer, H. W., Osinga, H. M. and Vegter, G. (1997) Algorithms for computing normally hyperbolic invariant manifolds, Z. Angew. Math. Phys. 48, 480-524.
8. Broer, H. W., Simó, C. and Tatjer, J. C. (1998) Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Nonlinearity 11, 667-770.
9. Chow, S.-N. and Hale, J. K. (1982) Methods of Bifurcation Theory, Springer-Verlag.
10. Fenichel, N. (1971) Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21, 193-225.
11. Fenichel, N. (1974) Asymptotic stability with rate conditions, Indiana Univ. Math. J. 23, 1109-1137.
12. Fenichel, N. (1977) Asymptotic stability with rate conditions, II, Indiana Univ. Math. J. 26, 81-93.
13. Fenichel, N. (1979) Geometric singular perturbations theory for ordinary differential equations, J. Differential Equations 31, 53-98.
14. Grasman, J. (1987) Asymptotic Methods for Relaxation Oscillations and Applications, Appl. Math. Sci. 63, Springer-Verlag.
15. Hale, J. K. (1963) Oscillations in Nonlinear Systems, MacGraw-Hill, New York, repr. Dover Publ., New York, 1992.
16. Hale, J. K. (1969) Ordinary Differential Equations, Wiley-Interscience.
17. Hirsch, M., Pugh, C. and Shub, M. (1977) Invariant Manifolds, Lecture Notes in Math. 583, Springer-Verlag, Heidelberg.
18. Jones, C. K. R. T. (1994) Geometric singular perturbation theory, In: R. Johnson (ed.), Dynamical Systems (Montecatini Terme, 1994), Lecture Notes in Math. 1609, Springer-Verlag, pp. 44-118.
19. Kaper, T. J. (1999) An introduction to geometric methods and dynamical systems theory for singular perturbation problems, In: J. Cronin and R. E. O'Malley, Jr. (eds), Proc. Symposia Appl. Math. AMS 56: Analyzing Multiscale Phenomena Using Singular Perturbation Methods, pp. 85-131.
20. Kaper, T. J. and Jones, C. K. R. T. (2001) A primer on the exchange lemma for fast-slow systems, In: C. K. R. T. Jones and A. I. Khibnik (eds), IMA Volumes in Mathematics and its Applications 122: Multiple-Time-Scale Dynamical Systems, Springer-Verlag, New York.
21. Kuznetsov, Yu. A. (2004) Elements of Applied Bifurcation Theory, 3rd edn, Springer-Verlag.
22. Schilder, F., Osinga, H. M. and Vogt, W. (2004) Continuation of quasi-periodic invariant tori, Preprint, Bristol (see also the site www.enm.bris.ac.uk/anm/preprints).
23. Shub, M. (1987) Global Stability of Dynamical Systems, Springer-Verlag.
24. Verhulst, F. (2000) Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer-Verlag.
25. Verhulst, F. (2005) Methods and Applications of Singular Perturbations, Springer-Verlag.
26. Verhulst, F. and Abadi (2005) Autoparametric resonance of relaxation oscillations, ZAMM 85, 122-131.