Kneadings, symbolic dynamics and painting lorenz chaos

International Journal of Bifurcation and Chaos

Volumn 22, Issue 4, 2012, Pages

  Barrio, Roberto ^a   Shilnikov, Andrey ^b   Shilnikov, Leonid ^c  

^a UNIVERSITY OF ZARAGOZA   (Spain)

^b GEORGIA STATE UNIVERSITY   (United States)

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

Author keywords

chaos; homoclinic and heteroclinic orbits; Kneading invariant; Lorenz attractor; symbolic dynamics; T points

Indexed keywords

CHAOS THEORY; NONLINEAR OPTICS;

HETEROCLINIC ORBIT; KNEADING INVARIANT; LORENZ ATTRACTOR; SYMBOLIC DYNAMICS; T-POINTS;

NONLINEAR EQUATIONS;

EID: 84861207436     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127412300169     Document Type: Review

References (62)

1. Abad, A., Barrio, R., Blesa, F & Rodŕ?guez, M. [2011a] "TIDES, a Taylor integrator for differential equations," ACM Trans Math. Softw., in press
2. Abad, A., Barrio, R., Blesa, F & Rodŕ?guez, M. [2011b] TIDES web page: http://gme.unizar.es/software/ tides
3. Afraimovich, V., Bykov, V. V & Shilnikov, L. P. [1977] "The origin and structure of the Lorenz attractor," Sov. Phys. Dokl. 22, 253-255
4. Afraimovich, V S & Shilnikov, L. P. [1983] "Strange attractors and quasiattractors," Nonlinear Dynamics and Turbulence, Interaction Mech. Math. Ser. (Pitman, Boston, MA), pp. 1-34
5. Afraimovich, V., Bykov, V. V & Shilnikov, L. P. [1983] "On structurally unstable attracting limit sets of Lorenz attractor type," Trans. Moscow Math. Soc. 44, 153-216
6. Barrio, R & Serrano, S. [2007] "A three-parametric study of the Lorenz model," Physica D 229, 43-51
7. Barrio, R & Serrano, S. [2009] "Bounds for the chaotic region in the Lorenz model," Physica D 238, 1615-1624
8. Barrio, R., Blesa, F & Serrano, S. [2009] "Qualitative analysis of the Rössler equations: Bifurcations of limit cycles and chaotic attractors," Physica D 238, 1087-1100
9. Barrio, R., Blesa, F & Serrano, S. [2011a] "Behavior patterns in multiparametric dynamical systems: Lorenz model," Int. J. Bifurcation and Chaos, in press
10. Barrio, R., Blesa, F., Serrano, S & Shilnikov, A. [2011b] "Global organization of spiral structures in biparameter space of dissipative systems with Shilnikov saddle-foci," Phys. Rev. E 84, 035201
11. Barrio, R., Rodŕ?guez, M., Abad, A & Blesa, F. [2011c] "Breaking the limits: The Taylor series method," Appl. Math. Comput. 217, 7940-7954
12. Bykov, V. [1998] "In bifurcations leasing to chaos in Chua's circuit," Int. J. Bifurcation and Chaos 8, 685-699
13. Bykov V. V, Shilnikov A 1989 Boundaries of the domain of existence of a Lorenz attractor Methods in Qualitative Theory and Bifurcation Theory 151-159 (in Russian)
14. Bykov V. V, Shilnikov A. L 1992 On the boundaries of the domain of existence of the Lorenz attractor Selecta Math. Sov. 11 375-382
15. Bykov V. V 1993 The bifurcations of separatrix contours and chaos Physica D 62 290-299
16. Fernández-Sánchez, F., Freire, E & Rodŕ?guez- Luis, A. J. [2002] "T-points in a Z2-symmetric electronic oscillator. I. Analysis," Nonlin. Dyn. 28, 53-69
17. Forysiak, W., Moloney, J & Harrison, R. [1991] "Bifurcations of an optically pumped three-level laser model," Physica D 53, 162-186
18. Gallas, J. A. C. [2010] "The structure of infinite periodic and chaotic hub cascades in phase diagrams of simple autonomous flows," Int. J. Bifurcation and Chaos 20, 197-211
19. Gaspard, P., Kapral, R & Nicolis, G. [1984] "Bifurcation phenomena near homoclinic systems: A twoparameter analysis," J. Stat. Phys. 35, 697-727
20. Glendinning, P & Sparrow, C. [1984] "Local and global behavior near homoclinic orbits," J. Stat. Phys. 35, 645-696
21. Glendinning, P & Sparrow, C. [1986] "T-points: A codimension two heteroclinic bifurcation," J. Stat. Phys. 43, 479-488
22. Glendinning, P & Sparrow, C. [1993] "Prime and renormalisable kneading invariants and the dynamics of expanding Lorenz maps," Physica D 1-4, 22-50
23. Glendinning, P & Hall, T. [1996] "Zeros of the kneading invariant and topological entropy for Lorenz maps," Nonlinearity 9, 999-1014
24. Gonchenko, S., Ovsyannikov, I., Simo, C & Turaev, D. [2005] "Three-dimensional Hénon-like maps and wild Lorenz-like attractors," Int. J. Bifurcation and Chaos 15, 3493-3508
25. Gonchenko, S. V., Shil'nikov, L. P & Turaev, D. V. [1993] "On models with non-rough Poincaré homoclinic curves," Physica D 1-4, 1-14
26. Gonchenko, S. V., Shil'nikov, L. P & Turaev, D. V. [1996] "Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits," Chaos 6, 15-31
27. Guckenheimer, J & Williams, R. F. [1979] "Structural stability of Lorenz attractors," Inst. Hautes Etudes Sci. Publ. Math. 50, 59-72
28. Haken, H. [1975] "Analogy between higher instabilities in fluids and lasers," Phys. Lett. A 53, 77-78
29. Hubner, U., Weiss, C., Abraham, N & Tang, D. [1995] "Lorenz-like chaos in nh3-fir lasers," Infr. Phys. Technol. 36, 489-512
30. Kaplan, J. L & Yorke, J. A. [1979] "Preturbulence: A regime observed in a fluid flow model of Lorenz," Commun. Math. Phys. 67, 93-108
31. Li, M.-C & Malkin, M. [2003] "Smooth symmetric and Lorenz models for unimodal maps," Int. J. Bifurcation and Chaos 13, 3353-3371
32. Lorenz, E. [1963] "Deterministic nonperiodic flow," J. Atmos. Sci. 20, 130-141
33. Lorenz, E. N. [2008] "Compound windows of the Hénonmap," Physica D 237, 1689-1704
34. Malkin, M. [1991] "Rotation intervals and dynamics of Lorenz type mappings," Selecta Math. Sov. 10, 265-275
35. Milnor, J & Thurston, W. [1988] "On iterated maps of the interval," Lecture Notes in Mathematics, Vol. 1342, pp. 465-563
36. Moloney, J., Forysiak, W., Uppal, T & Harrison, R. [1989] "Regular and chaotic dynamics of optically pumped molecular lasers," Phys. Rev. A 39, 1277-1285
37. Pertovksaya, N & Yudovich, V. [1980] "Homolinic loops on the Zaltzman-Lorenz system," Methods of Qualitative Theory of Differential Equations, Gorky University, pp. 73-83
38. Rand, D. [1978] "The topological classification of Lorenz attractors," Math. Proc. Camb. Phil. Soc. 83, 451-460
39. Robinson, C. [1989] "Homoclinic bifurcation a transitive attractor of Lorenz type," Nonlinearity 2, 495-518
40. Roshchin, N. [1978] "Dangerous stability boundaries in Lorenz's model," Prikl. Mat. Mekhanika 42, 950
41. Rychlic, M. [1990] "Lorenz attractor through Shil'nikov type bifurcation I," Erof. Th. Dyn. Syst. 10, 793-821
42. Shilnikov, A. [1986] "Bifurcations and chaos in the Morioka-Shimizu model Part I," Methods in Qualitative Theory and Bifurcation Theory, pp. 180-193 (in Russian)
43. Shilnikov, A. [1989] "Bifurcations and chaos in the Morioka-Shimizu model Part II," Methods in Qualitative Theory and Bifurcation Theory, pp. 130-138 (in Russian)
44. Shilnikov, A. [1991] "Bifurcation and chaos in the Morioka-Shimizu system," Selecta Math Sov. 10, 105-117
45. Shilnikov, A. [1993] "On bifurcations of the Lorenz attractor in the Shimizu-Morioka model," Physica D 62, 338-346
46. Shilnikov, A & Shilnikov, L. [1991] "On the nonsymmetric Lorenz model," Int. J. Bifurcation and Chaos 1, 773-776
47. Shilnikov, A. L., Shilnikov, L. P Turaev, D. V. [1993] "Normal forms and Lorenz attractors," Int. J. Bifurcation and Chaos 3, 1123-1139
48. Shilnikov, L. [1967] "The existence of a denumerable set of periodic motions in four-dimensional space in an extended neighborhood of a saddle-focus," Sov. Math. Dokl. 8, 54-58
49. Shilnikov, L. [1968] "On the birth of a periodic motion from a trajectory bi-asymptotic to an equilibrium state PF the saddle type," Sov. Math. Sbornik. 35, 240-264
50. Shilnikov, L. [1980] "Bifurcation theory and the Lorenz model," Appendix to Russian edition of "The Hopf Bifurcation and Its Applications", eds. Marsden, J. and McCraken, M., pp. 317-335
51. Shilnikov, L. [1981] "The theory of bifurcations and quasiattractors," Uspeh. Math. Nauk 36, 240-242
52. Shilnikov, L. [1994] "Chua's circuit: Rigorous results and future problems," Int. J. Bifurcation and Chaos 4, 489-518
53. Shilnikov, L. [1997] "Mathematical problems of nonlinear dynamics: A tutorial. Visions of nonlinear mechanics in the 21st century," J. Frank. Inst. 334, 793-864
54. Shilnikov, L. [2002] "Bifurcations and strange attractors," Proc. Int. Congr. Mathematicians (Beijing, China) (Invited Lectures) 3, 349-372
55. Shilnikov, L & Shilnikov, A. [2007] "Shilnikov bifurcation," Scholarpedia 2, 1891.
56. Shilnikov, L. P. [1965] "A case of the existence of a countable number of periodic motions," Sov. Math. Dokl. 6, 163
57. Shilnikov, L. P., Shilnikov, A. L., Turaev, D & Chua, L. O. [1998, 2001] Methods of Qualitative Theory in Nonlinear Dynamics. Parts I and II (World Scientific, Singapore)
58. Shimizu, T & Morioka, N. [1980] "On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model," Phys. Lett. A 76, 201-204
59. Sparrow, C. [1982] The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Applied Mathematical Sciences, Vol. 41 (Springer-Verlag, NY)
60. Tresser, C & Williams, R. [1993] "Splitting words and Lorenz braids," Physica D 62, 15-21
61. Turaev, D & Shilnikov, L. [1998] "An example of a wild strange attractor," Sbornik. Math. 189, 291-314
62. Vladimirov, A & Volkov, D. [1993] "Low-intensity chaotic operations of a laser with a saturable absorber," Opt. Commun. 100, 351-360.