Three-dimensional Hénon-like maps and wild Lorenz-like attractors

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 15, Issue 11, 2005, Pages 3493-3508

  Gonchenko, S V ^a   Ovsyannikov, I I ^a   Simo C ^b   Turaev, D ^c  

^a LOBACHEVSKY STATE UNIVERSITY OF NIZHNY NOVGOROD   (Russian Federation)

^b UNIVERSITY OF BARCELONA   (Spain)

^c BEN GURION UNIVERSITY OF THE NEGEV   (Israel)

Author keywords

Bifurcation; Lorenz attractor; Lyapunov exponent; Strange attractor

Indexed keywords

EID: 32044445484     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127405014180     Document Type: Article

References (51)

1. Afraimovich, V. S. & Shilnikov, L. P. [1974] "On some global bifurcations connected with the disappearance of a fixed point of saddle-node type," Sov. Math. Dokl. 15, 1761-1765.
2. Afraimovich, V. S., Bykov, V. V. & Shilnikov, L. P. [1977] "The origin and structure of the Lorenz attractor," Sov. Phys. Dokl. 22, 253-255.
3. Afraimovich, V. S., Bykov, V. V. & Shilnikov, L. P. [1980] "On the existence of stable periodic orbits in the Lorenz model," Russ. Math. Surv. 36.
4. Afraimovich, V. S. & Shilnikov, L. P. [1982] "Bifurcation of codimension 1, leading to the appearance of a countable set of tori," Sov. Math. Dokl. 25, 101-105.
5. Afraimovich, V. S., Bykov, V. V. & Shilnikov, L. P. [1983] "On attracting structurally unstable limit sets of Lorenz attractor type," Trans. Mosc. Math. Soc. 44, 153-216.
6. Afraimovich, V. S. & Shilnikov, L. P. [1983a] "Invariant two-dimensional tori, their breakdown and stochasticity," in Methods of the Qualitative Theory of Differential Equations (Gorki), pp. 3-26;
7. English translation [1991] in Transl, Ser. 2, Am. Math. Soc. 149, 201-212.
8. Afraimovich, V. S. & Shilnikov, L. P. [1983b] "Strange attractors and quasi-attractors," in Nonlinear Dynamics and Turbulence, Interaction Mech. Math. Ser. (Pitman), pp. 1-34.
9. Arneodo, A., Coullet, P. H. & Spiegel, E. A. [1985a] "The dynamics of triple convection," Geophys. Astrophys. Fluid Dyn. 31, 1-48.
10. Arneodo, A., Coullet, P. H., Spiegel, E. A. & Tresser, C. [1985b] "Asymptotic chaos," Physica D14, 327-347.
11. Belykh, V. N. [1984] "Bifurcation of separatrices of a saddle point of the Lorentz system," Diff. Eqs. 20, 1184-1191.
12. Bosch, M., Carcassés, J. P., Mira, C., Simó, C. & Tatjer, J. C. [1991a] "Crossroad area-spring area transition. (I) Parameter plane representation," Int. J. Bifurcation and Chaos 1, 183-196.
13. Bosch, M., Carcassés, J. P., Mira, C., Simó, C. & Tatjer, J. C. [1991b] "Crossroad area-spring area transition. (I) Foliated parametric representation," Int. J. Bifurcation and Chaos 1, 339-348.
14. Broer, H., Roussarie, R. & Simó, C. [1996] "Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphisms," Ergod. Th. Dyn. Syst. 16, 1147-1172.
15. Broer, H., Simó, C. & Tatjer, J. C. [1998] "Towards global models near homoclinic tangencies of dissipative diffeomorphisms," Nonlinearity 11, 667-770.
16. Broer, H., Simó, C. & Vitolo, R. [2005] "Chaos and quasi-periodicity in diffeomorphisms of the solid torus," in preparation.
17. Bykov, V. V. [1980] "Bifurcations of dynamical systems close to systems with a separatrix contour containing a saddle-focus," in Methods of the Qualitative Theory of Differential Equations (Gorky), pp. 44-72.
18. Bykov, V. V. & Shilnikov, A. L. [1989] "On the boundaries of the domain of existence of the Lorenz attractor," in Methods of Qualitative Theory and Bifurcation Theory (Gorky), pp. 151-159;
19. English translation [1992] in Selecta Math. Sov. 11, 375-382.
20. Bykov, V. V. [1993] "The bifurcations of separatrix contours and chaos," Physica D62, 290-299.
21. Fenichel, N. [1971] "Persistence and smoothness of invariant manifolds for flows," Indiana Univ. Math. J. 21, 193-226.
22. Gonchenko, S. V., Turaev, D. V. & Shilnikov, L. P. [1993] "Dynamical phenomena in multi-dimensional systems with a structurally unstable homoclinic Poincaré curve," Russ. Acad. Sci. Dokl. Math. 47, 410-415.
23. Gonchenko, S. V., Turaev, D. V. & Shilnikov, L. P. [2003] "On dynamical properties of diffeomorphisms with homoclinic tangencies," Contemp. Math. Appl. 7, 92-118; English version [2002] WIAS Preprint No.795.
24. Gonchenko, S. V., Meiss, J. D. & Ovsyannikov, I. I. [2006] "Chaotic dynamics of three-dimensional Hénon maps orginating from a homoclinic bifurcation," Int. Sci. J. Reg. Chaotic Dyn., to appear.
25. Gorodetskij, A. S. & Ilyashenko, Yu. S. [1999] "Some new robust properties of invariant sets and attractors of dynamical systems," Funct. Anal. Appl. 33, 95-105.
26. Gorodetskij, A. S. & Ilyashenko, Yu. S. [2000] "Some properties of skew products over a horseshoe and a solenoid," Proc. Steklov Inst. Math. 4, 90-112.
27. Gorodetskij, A., Ilyashenko, Yu., Kleptsyn, V. & Nalsky, M. [2005] "Non-removable zero Lyapunov exponents," Fund. Anal. Appl. 39.
28. Guckenheimer, J. [1976] "A strange, strange attractor," in The Hopf Bifurcation and its Application, eds. Marsden, J. E. & McCracen, M. (Springer-Verlag), pp. 368-381.
29. Guckenheimer, J. & Williams, R. F. [1979] "Structural stability of Lorenz attractors," Publ. Math. IHES 50, 59-71.
30. Lorenz, E. [1963] "Deterministic non-periodic flow," J. Atmos. Sci. 20, 130-141.
31. Neishtadt, A. [1984] "The separation of motions in systems with rapidly rotating phase," J. Appl. Math. Mech. 48, 133-139.
32. Ovsyannikov, I. M. & Shilnikov, L. P. [1987] "On systems with a saddle-focus homoclinic curve," Math. USSR Sb. 58, 557-574.
33. Ovsyannikov, I. M. & Shilnikov, L. P. [1992] "Systems with a homocline curve of a multidimensional saddle-focus, and spiral chaos," Math. USSR Sb. 73, 415-443.
34. Pisarevsky, V., Shilnikov, A. & Turaev, D. [1998] "Asymptotic normal forms for equilibria with a triplet of zero characteristic exponents in systems with symmetry," Reg. Chaotic Dyn. 2, 123-135.
35. Robinson, C. [1989] "Homoclinic bifurcation to a transitive attractor of Lorenz type," Nonlinearity 2, 495-518.
36. Robinson, C. [1992] "Homoclinic bifurcation to a transitive attractor of Lorenz type. II," SIAM J. Math. Anal. 23, 1255-1268.
37. Rychlik, M. R. [1990] "Lorenz attractors through Shilnikov-type bifurcation. I," Ergod. Th. Dyn. Syst. 10, 793-821.
38. Sataev, E. A. [2005] "Nonautonomous perturbations of systems of Lorenz type do not lead to the emergence of stable periodic orbits," in preparation.
39. Shilnikov, A. L. [1986] "Bifurcation and chaos in the Morioka-Shimizu system," in Methods of Qualitative Theory of Differential Equations (Gorky), pp. 180-193;
40. English translation [1991] in Selecta Math. Soviet. 10, 105-117.
41. Shilnikov, A. L. [1993] "On bifurcations of the Lorenz attractor in the Shimizu-Morioka model," Physica D62, 338-346.
42. Shilnikov, A. L., Shilnikov, L. P. & Turaev, D. V. [1993] "Normal forms and Lorenz attractors," Int. J. Bifurcation and Chaos 3, 1123-1139.
43. Shilnikov, A. L., Shilnikov, L. P. & Turaev, D. V. [2004] "On some mathematical topics in classical synchronization," Int. J. Bifurcation and Chaos 14, 2143-2160.
44. Shilnikov, L. P. [1969] "On the question of the structure of the neighborhood of a homoclinic tube of an invariant torus," Sov. Math. Dokl. 9, 624-628.
45. Shilnikov, L. P. [1981] "Bifurcation theory and quasi-hyperbolic attractors," Russ. Math. Surv. 36(4).
46. Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V. & Chua, L. O. [1998] Methods of Qualitative Theory in Nonlinear Dynamics. Part I (World Scientific, Singapore).
47. Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V. & Chua, L. O. [2001] Methods of Qualitative Theory in Nonlinear Dynamics. Part II (World Scientific, Singapore).
48. Shimizu, T. & Morioka, N. [1980] "On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model," Phys. Lett. A76, 201-204.
49. Tucker, W. [1999] "The Lorenz attractor exists," C. R. Acad. Sci. Paris Sér. I Math. 329, 1197-1202.
50. Turaev, D. V. & Shilnikov, L. P. [1998] "An example of a wild strange attractor," Sb. Math. 189, 137-160.
51. Turaev, D. V. & Shilnikov, L. P. [2005] "The Lorenz attractor under a small periodic forcing," in preparation.