Nonorientable manifolds in three-dimensional vector fields

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 13, Issue 3, 2003, Pages 553-570

  Osinga, Hinke M ^a  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

Bifurcation; M bius strip; Nonorientable invariant manifold; Vector field

Indexed keywords

ALGORITHMS; BIFURCATION (MATHEMATICS); COMPUTATIONAL METHODS; INVARIANCE;

THREE-DIMENSIONAL VECTOR FIELDS;

VECTORS;

EID: 0037986553     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127403006777     Document Type: Article

References (22)

1. Arneodo, A., Coullet, P. H., Spiegel, E. A. & Tresser, C. [1985] "Asymptotic chaos," Physica D14, 327-347.
2. Aronson, D. G., Chory, M. A., Hall, G. R. & McGehee, R. P. [1982] "Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study," Commun. Math. Phys. 83, 303-354.
3. Broer, H. W., Osinga, H. M. & Vegter, G. [1997] "Algorithms for computing normally hyperbolic invariant manifolds," Z. Angew. Math. Phys. 48(3), 480-524.
4. Escher, M. C. [2000] The Graphic Work (Benedikt Taschen Verlag, Köln).
5. Fenichel, N. [1971] "Persistence and smoothness of invariant manifolds for flows," Ind. Univ. Math. J. 21, 193-225.
6. Glendinning, P. & Sparrow, C. [1984] "Local and global behavior near homoclinic orbits," J. Stat. Phys. 35(5&6), 645-696.
7. Hénon, M. [1976] "A two-dimensional mapping with a strange attractor," Commun. Math. Phys. 50, 69-77.
8. Ilyashenko, Yu. S. & Li, W. [1999] Nonlocal Bifurcations, Mathematical Surveys and Monographs, Vol. 66 (American Mathematical Society, Providence, RI).
9. Johnson, M. E., Jolly, M. S. & Kevrekidis, I. G. [1997] "Two-dimensional invariant manifolds and global bifurcations: Some approximation and visualization studies," Num. Alg. 14, 125-140.
10. Krauskopf, B. & Osinga, H. M. [1998] "Growing 1d and quasi 2d unstable manifolds of maps," J. Comput. Phys. 146(1), 404-419.
11. Krauskopf, B. & Osinga, H. M. [1999] "Two-dimensional global manifolds of vector fields," Chaos 9(3), 768-774.
12. Krauskopf, B. & Osinga, H. M. [2000] "Investigating torus bifurcations in the forced Van der Pol oscillator," in Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, IMA Volumes in Mathematics and its Applications, Vol. 119, eds. Doedel, E. J. & Tuckerman, L. S. (Springer-Verlag, NY), pp. 199-208.
13. Kuznetsov, Yu. A. [1995] Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, Vol. 112, 2nd edition (Springer-Verlag, NY).
14. Lorenz, E. N. [1963] "Deterministic nonperiodic flow," J. Atmos. Sci. 20, 130-141.
15. Oldeman, B. E., Krauskopf, B. & Champneys, A. R. [2000] "Death of period-doublings: Locating the homoclinic-doubling cascade," Physica D146, 100-120.
16. Osinga, H. M. [2000] "Nonorientable manifolds of periodic orbits," Proc. Int. Conf. Differential Equations, Equadiff 99 (Berlin] Vol. 2, eds. Fiedler, B., Gröger, K. & Sprekels, J. (World Scientific, Singapore), pp. 922-924.
17. 3-model and in Sandstede's model; the website can be found at http://www.enm.bris.ac.uk/staff/hinke/nonorientable/.
18. Van der Pol, B. [1927] "Forced oscillations in a circuit with nonlinear resistance," London, Edinburgh and Dublin Phil. Mag. 3, 65-80.
19. Sandstede, B. [1997] "Constructing dynamical systems having homoclinic bifurcation points of codimension two," J. Dyn. Diff. Eqs. 9(2), 269-288.
20. Sotomayor, J. [1973] "Generic bifurcations of dynamical systems," in Dynamical Systems Proc. Symp. 1971, University of Bahia, Salvador, Brazil, ed. Peixoto, M. M. (Academic Press, NY), pp. 561-582.
21. Spivak, M. [1965] Calculus on Manifolds (Addison-Wesley, NY).
22. Wieczorek, S., Krauskopf, B. & Lenstra, D. [1999] "A unifying view of bifurcations in a semiconductor lases subject to optical injection," Opt. Commun. 172, 279-295.