Hyperbolicity and invariant manifolds for planar nonautonomous systems on finite time intervals

International Journal of Bifurcation and Chaos

Volumn 18, Issue 3, 2008, Pages 641-674

  Duc, Luu Hoang ^a   Siegmund, Stefan ^a  

^a GOETHE UNIVERSITY   (Germany)

Author keywords

Hyperbolicity; Invariant manifolds; Nonautonomous dynamical systems on finite time intervals; OkuboWeiss criterion

Indexed keywords

ASYMPTOTIC STABILITY; VORTEX FLOW;

AUTONOMOUS EQUATIONS; FINITE TIME INTERVALS; GRAPH TRANSFORMATION; HYPERBOLICITY; INVARIANT MANIFOLDS; NON-AUTONOMOUS SYSTEM; NONAUTONOMOUS DYNAMICAL SYSTEMS; OKUBOWEISS CRITERION;

DYNAMICAL SYSTEMS;

EID: 45149090989     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127408020562     Document Type: Article

References (45)

1. Barbǎalat, I. [1959] "Systèmes d'équations différentielles d'oscillations nonlinéaires," Revue de Mathématiques Pures et Appliquées, Bucharest IV, 267-270.
2. Barreira, L. & Valls, C. [2005] "Higher regularity of invariant manifolds for nonautonomous equations," Nonlinearity 18, 2373-2390.
3. Batchelor, G. K. [1967] An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge).
4. Berger, A., Doan, T. S. & Siegmund, S. [2008] "Nonautonomous finite-time dynamics," Discr. Contin. Dyn. Syst. Series B 9, 463-492.
5. Berger, A. & Siegmund, S. [2003] "On the gap between random dynamical systems and continuous skew products," J. Dyn. Diff. Eqs. 15, 237-263.
6. Chow, S.-N., Li, C. & Wang, D. [1994] Normal Forms and Bifurcation of Planar Vector Fields (Cambrigde University Press, Cambridge).
7. Conley, C. [1978] Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, Vol. 38 (American Mathematical Society, Providence, RI).
8. Dresselhaus, E. & Tabor, M. [1992] "The kinematics of stretching and alignment of material elements in general flow fields," J. Fluid Mech. 236, 415-444.
9. Duc, L. H. [2006] Finite Time Dynamics versus Asymptotic Dynamics for Nonautonomous Systems (Dissertation, University of Frankfurt).
10. Duc, L. H. & Siegmund, S. [2008] A Bifurcation Model for Symmetric Two-Dimensional Vortex Merger, submitted.
11. Frisius, T., Hasselbeck, T. & Herbert, F. [2006] "The effect of ice-phase microphysics on tropical cyclones simulated by the Lokalmodell," COSMO Newslett. 6, 236-241.
12. Hale, J. [1980] Ordinary Differential Equations, Second edition (Robert E. Krieger Publishing Co., Inc., Huntington, NY).
13. Haller, G. [2000] "Finding finite-time invariant manifolds in two-dimensional velocity fields," Chaos 10, 99-108.
14. Haller, G. [2001a] "Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence," Phys. Fluids 13, 3365-3385.
15. Haller, G. [2001b] "Distinguished material surfaces and coherent structures in 3D fluid flows," Physica D 149, 248-277.
16. Haller, G. [2002] "Lagrangian coherent structures from approximate velocity data," Phys. Fluids A 14, 1851-1861.
17. Haller, G. [2005] "An objective definition of a vortex," J. Fluid Mech. 525, 1-26.
18. Haller, G. & Poje, A. C. [1998] "Finite-time transport in aperiodic flows," Physica D 119, 352-380.
19. Haller, G. & Yuan, G. [2000] "Lagrangian coherent structures and mixing in two-dimensional turbulence," Physica D 147, 352-370.
20. Henry, D. [1981] Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840 (Springer-Verlag, Berlin-NY).
21. Hua, B. L. & Klein, P. [1998] "An exact criterion for the stirring properties of nearly two-dimensional turbulence," Phys. D 113, 98-110.
22. Ide, K., Small, D. & Wiggins, S. [2002] "Distinguished hyperbolic trajectories in time-dependent fluid flows: Analytical and computational approach for velocity fields defined as data sets," Nonlin. Process. Geophys. 9, 237-263.
23. Ju, N., Small, D. & Wiggins, S. [2002] "Existence and computation of hyperbolic trajectories of aperiodically time dependent vector fields and their approximations," Int. J. Bifurcation and Chaos 13, 1449-1457.
24. Keller, H. [1977] "Numerical solution of bifurcation and nonlinear eigenvalue problems," Applications of Bifurcation Theory, Publ. Math. Res. Center, Vol. 38 (Academic Press, NY), pp. 359-384.
25. Koh, T.-Y. & Legras, B. [2002] "Hyperbolic lines and the stratospheric polar vortex," Chaos 12, 382-394.
26. Kloeden, P. E. & Siegmund, S. [2005] "Bifurcations and continuous transitions of attractors in autonomous and nonautonomous systems," Int. J. Bifurcation and Chaos 15, 743-762.
27. Langa, J. & Robinson, J. & Suárez, A. [2002] "Stability, instability, and bifurcation phenomena in nonautonomous differential equations," Nonlinearity 15, 887-903.
28. Lekien, F., Shadden, S. C. & Marsden, J. E. [2006] "N-Dimensional Lagrangian coherent structures: Definition and properties," Physica D, in preparation.
29. Malhotra, N. & Wiggins, S. [1998] "Geometric structures, lobe dynamics, and Lagrangian transport in flows with aperiodic time-dependence, with applications to Rossby wave flow," J. Nonlin. Sci. 8, 401-456.
30. Mancho, A., Small, D., Wiggins, S. & Ide, K. [2003] "Computation of stable and unstable manifolds of hyperbolic trajectories in two-dimensional, aperiodically time-dependent vector fields," Physica D 182, 188-222.
31. Mancho, A., Small, D. & Wiggins, S. [2004] "Computation of hyperbolic trajectories and their stable and unstable manifolds for oceanographic flows represented as data sets," Nonlin. Process. Geophys. 11, 17-33.
32. Miller, P. D., Jones, C. K. R. T., Rogerson, A. M. & Pratt, L. J. [1997] "Quantifying transport in numerically generated velocity fields," Physica D 110, 105-122.
33. Mischaikow, K. [1999] The Conley Index Theory: A Brief Introduction, Banach Center Publ., Vol. 47 (Polish Acad. Sci., Warsaw).
34. Okubo, A. [1970] "Horizontal dispersion of floatable particles in the vicinity of velocity singularities such as convergences," Deep-Sea Res. 17, 445-454.
35. Ottino, J. [1989] The Kinematics of Mixing: Stretching, Chaos, and Transport (Cambridge University Press, Cambridge).
36. Rasmussen, M. [2007] Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Lecture Notes in Mathematics, Vol. 1907 (Springer-Verlag).
37. Sandstede, B., Balasuriya, S., Jones, C. K. R. T. & Miller, P. [2000] "Melnikov theory for finite-time vector fields," Nonlinearity 13, 1357-1377.
38. Sell, G. [1971] Topological Dynamics and Ordinary Differential Equations (Van Nostrand Reinhold Co., London).
39. Sell, G. & You, Y. [2002] Dynamics of Evolutionary Equations, Applied Mathematical Sciences, Vol. 143 (Springer, NY).
40. Shadden, S. C., Lekien, F. & Marsden, J. E. [2005] "Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows," Physica D 212, 271-304.
41. Tabor, M. & Klapper, I. [1994] "Stretching and alignment in chaotic and turbulent flows," Chaos Solit. Fract. 4, 1031-1055.
42. Velasco Fuentes, O. U. [2001] "Chaotic advection by two interacting fininte-area vortices," Phys. Fluids 13, 901-912.
43. Weiss, J. [1991] "The dynamics of enstrophy transfer in two-dimensional hydrodynamics," Physica D 48, 273-294.
44. Wiggins, S. [1992] Chaotic Transport in Dynamical Systems (Springer, NY).
45. Wiggins, S. [2005] "The dynamical systems approach to Lagrangian transport in Oceanic flows," Annu. Rev. Fluid Mech. 37, 295-328.