Strange attractors for periodically forced parabolic equations

Memoirs of the American Mathematical Society

Volumn 224, Issue 1054, 2013, Pages 1-97

  Lu, Kening ^a   Wang, Qiudong ^a   Young, Lai Sang ^a  

^a NONE

Author keywords

Hopf bifurcations; Parabolic PDEs; Periodic forcing; SRB measures; Strange attractors

Indexed keywords

EID: 84878812752     PISSN: 00659266     EISSN: 19476221     Source Type: Journal    
DOI: 10.1090/s0065-9266-2012-00669-1     Document Type: Article

References (56)

1. [A] A. Andronov. Application of Poincar'e's theorem on "bifurcation points" and "change in stability" to simple auto-oscillatory systems, C.R. Acad. Sci. (Paris), 189 (1929), 559-561.
2. [B] R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math. 470, Springer, Berlin, (1975). MR0442989 (56:1364
3. [BC] M. Benedicks, L. Carleson. The dynamics of the H́enon map, Ann. of Math. (2) 133(1) (1991), 73-169. MR1087346 (92d:58116
4. [BLZ] P. Bates, K. Lu, and C. Zeng. Approximately Invariant Manifolds and Global Dynamics of Spike States, Invent. Math. 174 (2008), 355-433. MR2439610 (2011b:37157
5. [BV] A.V. Babin and M.I. Vishik. Attractors of Evolution Equations, North-Holland, Amsterdam, (1992). MR1156492 (93d:58090
6. [BY] M. Benedicks and L.-S. Young. SBR measures for certain Henon maps, Inventiones Math. 112 (1993), 541-576. MR1218323 (94e:58074
7. [C] G. Carpenter. Bursting phenomena in excitable membranes, SIAM J. Appl. Math. 36(2) (1979), 334-372. MR524506 (80b:92007
8. [CLL] S-N. Chow, K. Lu, and X-B. Lin. Smooth foliations for flows in Banach space, Journal of Differential Equations, 94 (1991), 266-291. MR1137616 (92k:58210
9. [DPo] E. N. Dancer and P. Poĺacik. Realization of vector fields and dynamics of spatially homogeneous parabolic equations, Memoirs of Amer. Math. Society, 140 (1999), number 668, 82 pp. MR1618487 (99m:35125
10. [DRV] L.J. Diaz, J. Rocha and M. Viana. Strange attractors in saddle-node cycles: prevalence and globality, Invent. Math. 125 (1996), 37-74. MR1389960 (97h:58109
11. [G] J. Guckenheimer. On a codimension two bifurcation, Lecture Notes in Math., 898, Springer-Verlag, (1980), 99-142. MR654886 (83j:58088
12. [GH] J. Guckenheimer and P. Holmes. Nonlinear oscillators, dynamical systems and bifurcations of vector fields, Springer-Verlag, Appl. Math. Sciences, 42 (1983). MR709768 (85f:58002
13. [HKW] B. Hassard, N. Kazarinoff, and Y-H. Wan. Theory and Applications of Hopf Bifurcation, Cambridge University Press, (1981). MR603442 (82j:58089
14. [H] D. Henry. Geometric Theory of Semilinear Parabolic Equations, Springer, New York, (1981). MR610244 (83j:35084
15. [Ha] J.K. Hale. Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, (1988). MR941371 (89g:58059
16. [HL] J. Hale and X-B. Lin. Examples of transverse homoclinic orbits in delay equations, Nonlinear Anal., 10(7) (1986), 693-709. MR849959 (87i:34087
17. [HMO] J. Hale, L. Magalhaes and W. Oliva. Dynamics in Infinite Dimensions, Applied Math. Sc., Springer-Verkag, 47, (2002). MR1914080 (2004j:37001
18. [He] M. Herman. Mesure de Lebesgue et Nombre de Rotation, Lecture Notes in Math., Springer, 597, (1977), 271-293. MR0458480 (56:16682
19. [HM] P. Holmes and J. Marsden. A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam, Arch. Rational Mech. Anal., 76 (1981), 135-166. MR629701 (82m:58039
20. [Ho] E. Hopf. Abzweigung einer periodischen losung von einer stationären losung eines differentialsystems, Ber. Verh. Sächs, Acad. Wiss. Leipzig Math. Phys., 94 (1942), 1-22.
21. [HPS] M. Hirsch, C. Pugh and M. Shub. Invariant Manifolds, Lecture Notes in Math., Springer Verlag, 583 (1977). MR0501173 (58:18595
22. [J] M. Jakobson. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Commun. Math. Phys. 81 (1981), 39-88. MR630331 (83j:58070
23. [K] T. Kato. Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York (1966). MR0203473 (34:3324
24. [KR] N. Kopell and D. Ruelle. Bounds on complexity in reaction-diffusion systems, SIAM J. Appl. Math. 46 (1986), 68-80. MR821442 (87d:35020
25. [KTS] P. Kuznetsov, L. V. Turukina, and E. Mosekilde. Dynamical systems of different classes as models of the kicked nonlinear oscillator, A., International Journal of Bifurcation and Chaos, 11 (2001), 1065-1077. MR1839905 (2002b:37043
26. [LMSW] Y. Li, David W. McLaughlin, Jalal Shatah, and S. Wiggins. Persistent homoclinic orbits for a perturbed nonlinear Schrodinger equation, Comm. Pure Appl. Math., 49(11) (1996), 1175-1255. MR1406663 (98d:35208
27. [LY] F. Ledrappier and L.-S. Young. The metric entropy of diffeomorphisms, Annals of Math. (2) 122 (1985), 509-574. MR819556 (87i:58101a
28. [LP] R. Lefever and I. Prigogine. Symmetry-breaking instability in dissipative systems II, J. Chem. Phy., 48 (1968), 1695-1700.
29. [L] Y. Li. Smale horseshoes and symbolic dynamics in perturbed nonlinear Schrodinger equations, J. Nonlinear Sci., 9(4) (1999), 363-415. MR1700670 (2000m:37173
30. [M] M. Misiurewicz, Absolutely continuous invariant measures for certain maps of an interval, Publ. Math. IHES. 53 (1981), 17-51. MR623533 (83j:58072
31. [Mo] J. Moser, Stable and random motions in dynamical systems, Princeton University Press, Princeton, NJ, (1973). MR0442980 (56:1355
32. [MV] L. Mora and M. Viana, Abundance of strange attractors, Acta. Math. 171 (1993), 1-71. MR1237897 (94k:58089
33. [P] Ya. Pesin. Characteristic Lyapunov exponents and smooth ergodic theory, Russ. Math. Surveys 32 (1977), 55-114. MR0466791 (57:6667
34. [Po1] P. Pol'acik. Parabolic equations: asymptotic behavior and dynamics on invariant manifolds (survey). Handbook on Dynamical Systems vol. 2, B. Fiedler (ed.), Elsevier, Amsterdam (2002), 835-883. MR1901067 (2003f:37150
35. [Po2] P. Poĺacik. Realization of any finite jet in a scalar semilinear parabolic equation on the ball in R2 Annali Scuola Norm Pisa, 18 (1991), 83-102. MR1118222 (92g:35107
36. [Po3] P. Poĺacik. High dimensional ω-limit sets and chaos in scalar parabolic equations, Journal of Differential Equations, 119 (1995), 24-53. MR1334487 (96h:35092
37. [Po4] P. Poĺacik. Complicated dynamics in Scalar Semilinear Parabolic Equations, In Higher Space Dimensions, Journal of Differential Equations, 89 (1991), 244 -271. MR1091478 (92c:35063
38. [Ryb] K. P. Rybakowski. Realization of arbitrary vector fields on center manifolds of parabolic Dirichlet BVP's, J. Differential Equations, 114 (1994), 199-221. MR1302140 (95j:35114
39. [R1] D. Ruelle. A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976), 619-654. MR0415683 (54:3763
40. [R2] D. Ruelle. Ergodic theory of differentiable dynamical systems, Publ. Math., Inst. Hautes ́etud. Sci. 50 (1979), 27-58. MR556581 (81f:58031
41. [SZ] J. Shatah and C. Zeng. Homoclinic orbits for the perturbed sine-Gordon equation, Comm. Pure Appl. Math., 53(3) (2000), 283-299. MR1725609 (2000h:37133
42. [S] Y. G. Sinai. Gibbs measure in ergodic theory, Russian Math. Surveys 27 (1972), 21-69. MR0399421 (53:3265
43. [Sm] S. Smale. Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR0228014 (37:3598
44. [T] R. Temam. Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Math. Sc., Springer-Verlag, 68 (1997). MR1441312 (98b:58056
45. [Vak] S. A. Vakulenko. Dissipative systems generating any structurally stable chaos, Advances in Differential Equations, 5 (2000), 1139-1178. MR1776350 (2001k:37128
46. [VV] S. A. Vakulenko and V. Volpert. Generalized traveling waves for perturbed monotone reaction-diffusion systems, Nonlinear Anal. 46 (2001), Ser. A: Theory Methods, 757-776. MR1859795 (2002i:35105
47. [W] H-O. Walther. Homoclinic solution and chaos, Nonlinear Anal., 5(7) (1981), 775-788. MR623379 (83a:58060
48. [WO] Q. Wang and W. Ott. Dissipative homoclinic loops and rank one chaos, preprint (2007).
49. [WY1] Q. Wang and L.-S. Young. Strange attractors with one direction of instability, Commun. Math. Phys., 218 (2001), 1-97. MR1824198 (2002m:37050
50. [WY2] Q. Wang and L.-S. Young. From invariant curves to strange attractors, Commun. Math. Phys., 225 (2002), 275-304. MR1889226 (2003e:37045
51. [WY3] Q. Wang and L.-S. Young. Strange attractors in periodically-kicked limit cycles and Hopf bifurcations, Commun. Math. Phys., 240 (2002), 509-529. MR2005855 (2004i:37070
52. [WY4] Q. Wang and L.-S. Young. Toward a theory of rank one attractors, Ann. of Math. (2) 167, (2008), 349-480. MR2415378 (2009f:37036
53. [WY5] Q. Wang and L.-S. Young. Non-uniformly expanding 1D maps, Commun. Math. Phys, 264(1) (2006), 255-282. MR2212223 (2006k:37103
54. [Y1] L.-S. Young. Ergodic theory of differentiable dynamical systems, Real and Complex Dynamical Systems, (B. Branner and P. Hjorth, ed.), Kluwer Acad. Press, (1995), 293-336. MR1351527 (96h:58101
55. [Y2] L.-S. Young. What are SRB measures, and which dynamical systems have them?. J. Stat. Phys., 108 (2002), 733-754. MR1933431 (2003g:37042
56. [Z] C. Zeng. Homoclinic orbits for a perturbed nonlinear Schrodinger equation, Comm. Pure Appl. Math., 53(10) (2000), 1222-1283. MR1768814 (2002c:37126a)