All possible chaotic dynamics can be approximated in three dimensions

Nonlinearity

Volumn 11, Issue 3, 1998, Pages 529-545

  Robinson, James C ^a,b  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^b UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0007337855     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/11/3/007     Document Type: Article

References (29)

1. Billotti J E and LaSalle J P 1971 Dissipative periodic processes Bull. Am. Math. Soc. 77 1082-8
2. Bowen R 1975 ω-limit sets for Axiom A diffeomorphisms J. Diff. Equ. 18 333-9
3. Constantin P and Foias C 1988 Navier-Stokes Equations (Chicago: University of Chicago Press)
4. Constantin P, Foias C and Temam R 1985 Attractors representing turbulent flows Mem. Am. Math. Soc. 53
5. Coomes B A, Koçak H and Palmer K J 1995 Rigorous computational shadowing of orbits of ordinary differential equations Numer. Math. 69 401-21
6. Doering C R and Gibbon J D 1995 Applied Analysis of the Navier-Stokes Equations (Cambridge: Cambridge University Press)
7. Eden A, Foias C, Nicolaenko B and Temam R 1994 Exponential Attractors for Dissipative Evolution Equations (Chichester: Wiley)
8. Falconer K 1990 Fractal Geometry (Chichester: Wiley)
9. Foias C and Olson E J 1996 Finite fractal dimension and Hölder-Lipschitz parametrization Indiana Univ. Math. J. 45 603-16
10. Glendinning P 1988 Global bifurcations in flows New Directions in Dynamical Systems ed T Bedford and J Swift (Cambridge: Cambridge University Press) pp 120-49
11. Guillopé C 1982 Comportement à l'infini des solutions des équations de Navier-Stokes et propriété des ensembles fonctionnels invariantes (ou attracteurs) Ann. Inst. Fourier (Grenoble) 32 1-37
12. Hale J K 1988 Asymptotic Behavior of Dissipative Systems (Mathematical Surveys and Monographs 25) (Providence, RI: American Mathematical Society)
13. Hartman P 1964 Ordinary Differential Equations (New York: Wiley)
14. Heywood J G and Rannacher R (1982) Finite approximation of the nonstationary Navier-Stokes problem, Part I: Regularity of solutions and second-order error estimates for spatial discretization SIAM J. Numer. Anal. 19 275-311
15. Holmes P 1988 Knots and orbit genealogies in nonlinear oscillators New Directions in Dynamical Systems ed T Bedford and J Swift (Cambridge: Cambridge University Press) pp 150-91
16. Katok A and Hasselblatt B 1995 Introduction to the Modern Theory of Dynamical Systems (Cambridge: Cambridge University Press)
17. Langa J A and Robinson J C 1996 Determining asymptotic behaviour from the dynamics on attracting sets J. Dyn. Differ. Equ. (to appear)
18. Lorenz E 1963 Deterministic nonperiodic flow J. Atmos. Sci. 20 130-41
19. Mañé R 1981 On the dimension of the compact invariant sets of certain nonlinear maps Springer Lecture Notes in Mathematics vol 898 pp 230-42
20. Mielke A 1997 The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractors Nonlinearity 10 199-222
21. Robinson J C 1997a Global attractors: topology and finite-dimensional dynamics J. Dyn. Differ. Equ. (to appear)
22. Robinson J C 1997b Accurate pictures of any attractor using three ordinary differential equations Physica D (submitted)
23. Robinson J C 1998 All solutions of continuous ODEs can be obtained as the limit of solutions of Lipschitz ODEs (in preparation)
24. Rössler O E 1976 An equation for continuous chaos Phys. Lett. 57A 397-8
25. Sparrow C 1982 The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors (New York: Springer)
26. Stein E M 1970 Singular Integrals and Differentiability Properties of Functions (Princeton, NJ: Princeton University Press)
27. Stuart A M and Humphries A R 1996 Dynamical Systems and Numerical Analysis (Cambridge: Cambridge University Press)
28. Temam R 1979 Navier-Stokes Equations, Theory and Numerical Analysis (Amsterdam: North-Holland)
29. Temam R 1988 Infinite Dimensional Dynamical Systems in Mechanics and Physics (Applied Mathematics Sciences 68) (New York: Springer)