The many facets of chaos

International Journal of Bifurcation and Chaos

Volumn 25, Issue 4, 2015, Pages

  Sander, Evelyn ^a   Yorke, James A ^b  

^a GEORGE MASON UNIVERSITY   (United States)

^b Bldg 142   (United States)

Author keywords

basins of attraction; Chaos; forced damped pendulum; horseshoes; Lyapunov exponents

Indexed keywords

CHAOS THEORY; FRACTALS; LYAPUNOV METHODS;

BASIN OF ATTRACTION; BASINS OF ATTRACTION; DYNAMICAL BEHAVIORS; FORCED DAMPED PENDULUMS; HORSESHOES; LYAPUNOV EXPONENT; QUASI-PERIODIC ORBITS; SPECIFIC PROPERTIES;

DYNAMICAL SYSTEMS;

EID: 84929179953     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127415300116     Document Type: Article

References (49)

1. Arai, Z., Mischaikow, K. [2006] "Rigorous computations of homoclinic tangencies, " SIAM J. Appl. Dyn. Syst. 5, 280-292 (electronic).
2. Arnold, V. I. [2009] Vladimir I. Arnold-Collected Works. Representations of Functions, Celestial Mechanics and KAM Theory, 1957-1965, eds. Givental, A. B., Khesin, B. A., Marsden, J. E., Varchenko, A. N., Vassilev, V. A., Viro, O. Ya., Zakalyukin, V. M., Vol. I (Springer-Verlag, Berlin).
3. Barrio, R., Borczyk, W., Breiter, S. [2009] "Spurious structures in chaos indicators maps, " Chaos Solit. Fract. 40, 1697-1714.
4. Blanchard, F., Glasner, E., Kolyada, S., Maass, A. [2002] "On Li-Yorke pairs, " J. Reine Angew. Math. 547, 51-68.
5. Bonatti, C., Daz, L. J., Fisher, T. [2008] "Superexponential growth of the number of periodic orbits inside homoclinic classes, " Discr. Contin. Dyn. Syst. 20, 589-604.
6. Bowen, R. [1970] "Topological entropy and axiom A, " Global Analysis (Proc. Symp. Pure Math., Vol. XIV, Berkeley, Calif., 1968 ) (Amer. Math. Soc., Providence, RI), pp. 23-41.
7. Brown, M. [2010] How I Killed Pluto and Why It Had It Coming (Random House).
8. Chung, Y. M. [2001] "Topological entropy for differentiable maps of intervals, " Osaka J. Math. 38, 1-12.
9. Chung, Y. M., Hirayama, M. [2003] "Topological entropy and periodic orbits of saddle type for surface diffeomorphisms, " Hiroshima Math. J. 33, 189-195.
10. Day, S., Frongillo, R., Treviño, R. [2008] "Algorithms for rigorous entropy bounds and symbolic dynamics, " SIAM J. Appl. Dyn. Syst. 7, 1477-1506.
11. Devaney, R., Nitecki, Z. [1979] "Shift automorphisms in the Hénon mapping, " Comm. Math. Phys. 67, 137-146.
12. Frongillo, R., Treviño, R. [2012] "Efficient automation of index pairs in computational Conley index theory, " SIAM J. Appl. Dyn. Syst. 11, 82-109.
13. Gollub, J., Swinney, H. L. [1975] "Onset of turbulence in a rotating fluid, " Phys. Rev. Lett. 35, 927-930.
14. Gottwald, G. A., Melbourne, I. [2009a] "On the implementation of the 0-1 test for chaos, " SIAM J. Appl. Dyn. Syst. 8, 129-145.
15. Gottwald, G. A., Melbourne, I. [2009b] "On the validity of the 0-1 test for chaos, " Nonlinearity 22, 1367-1382.
16. Grebogi, C., Ott, E., Pelikan, S., Yorke, J. A. [1984] "Strange attractors that are not chaotic, " Physica D 13, 261-268.
17. Hayashi, C., Ueda, Y., Akamatsu, N., Itakura, H. [1970] "On the behavior of self-oscillatory systems with external force, " Trans. IECE 53-A, 150-158.
18. Hunt, B. R., Sauer, T., Yorke, J. A. [1992] "Prevalence: A translation-invariant "almost every" on infinitedimensional spaces, " Bull. Amer. Math. Soc. 27, 217-238.
19. Kaloshin, V. Y. [1997] "Prevalence in the space of finitely smooth maps, " Funct. Anal. Its Appl. 31, 95-99.
20. Kaloshin, V. Y., Hunt, B. R. [2001a] "A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms. I, " Electron. Res. Announc. Amer. Math. Soc. 7, 17-27 (electronic).
21. Kaloshin, V. Y., Hunt, B. R. [2001b] "A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms. II, " Electron. Res. Announc. Amer. Math. Soc. 7, 28-36 (electronic).
22. Kaloshin, V., Kozlovski, O. S. [2011] "A Cr unimodal map with an arbitrary fast growth of the number of periodic points, " Erg. Th. Dyn. Syst. 32, 159-165.
23. Katok, A., Mezhirov, A. [1998] "Entropy and growth of expanding periodic orbits for one-dimensional maps, " Fund. Math. 157, 245-254.
24. Lani-Wayda, B., Walther, H.-O. [1995] "Chaotic motion generated by delayed negative feedback. I. A transversality criterion, " Diff. Integr. Eqs. 8, 1407-1452.
25. Lani-Wayda, B., Walther, H.-O. [1996] "Chaotic motion generated by delayed negative feedback. II. Construction of nonlinearities, " Math. Nachr. 180, 141-211.
26. Lani-Wayda, B. [2001] "Wandering solutions of delay equations with sine-like feedback, " Mem. Amer. Math. Soc., Vol. 151, 121 pp.
27. Li, T., Yorke, J. A. [1975] "Period three implies chaos, " Amer. Math. Month. 82, 985-992.
28. Lorenz, E. N. [1963] "Deterministic nonperiodic flow, " J. Atmosph. Sci. 20, 130.
29. Melbourne, I., Gottwald, G. A. [2008] "Power spectra for deterministic chaotic dynamical systems, " Nonlinearity 21, 179.
30. Misiurewicz, M., Szlenk, W. [1980] "Entropy of piecewise monotone mappings, " Studia Math. 67, 45-63.
31. Mytkowicz, T., Diwan, A., Bradley, E. [2009] "Computer systems are dynamical systems, " Chaos 19, 033124.
32. Newhouse, S., Berz, M., Grote, J., Makino, K. [2008] "On the estimation of topological entropy on surfaces, " Geometric and Probabilistic Structures in Dynamics (Amer. Math. Soc., Providence, RI), pp. 243-270.
33. Ott, W., Yorke, J. A. [2005] "Prevalence, " Bull. Amer. Math. Soc. 42, 263-290.
34. Poincaré, H. [1993] New Methods of Celestial Mechanics, Vol. 13 (American Institute of Physics, Paris).
35. Sander, E., Yorke, J. [2009] "A classification of explosions in dimension one, " Erg. Th. Dyn. Syst. 29, 715-731.
36. Sander, E., Yorke, J. [2012] "Connecting perioddoubling cascades to chaos, " Int. J. Bifurcation and Chaos 22, 1250022-1-16.
37. Sander, E., Yorke, J. [2013] "A period-doubling cascade precedes chaos for planar maps, " Chaos 23, 033113.
38. Sander, E., Yorke, J. [2014] "Pendulum gyrations, " http://youtu.be/JyhaHyCvgw4, and http://math.gmu. edu/sander/EvelynSite/pendulum-gyrations.html.
39. Shadden, S. C. [2005] "Lagrangian coherent structures: Analysis of time-dependent dynamical systems using finite-time Lyapunov exponents, " DSWeb Tutorials Section. URL http://www.dynamicalsystems.org/tu/te/detailitem=27.
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. Slipantschuk, J., Bandtlow, O. F., Just, W. [2013] "On the relation between Lyapunov exponents and exponential decay of correlations, " J. Phys. A: Math. Theoret. 46, 075101.
42. Smale, S. [1967] "Differentiable dynamical systems, " Bull. Amer. Math. Soc. 73, 747-817.
43. Tempkin, J. A., Yorke, J. A. [2007] "Spurious Lyapunov exponents computed from data, " SIAM J. Appl. Dyn. Syst. 6, 457-474.
44. Ugarcovici, I., Weiss, H. [2004] "Chaotic dynamics of a nonlinear density dependent population model, " Nonlinearity 17, 1689-1711.
45. Wang, Q., Young, L. [1999] "Analysis of a class of strange attractors, " arXiv:math/9909012.
46. Wang, Q., Young, L.-S. [2001] "Strange attractors with one direction of instability, " Commun. Math. Phys. 218, 1-97.
47. Wang, Q., Young, L.-S. [2013] "Dynamical profile of a class of rank-one attractors, " Erg. Th. Dyn. Syst. 33, 1221-1264.
48. Weisstein, E. W. [2014] "Fermat's little theorem, " Math-World-A Wolfram Web Resource. URL http://mathworld.wolfram.com/FermatsLittleTheorem.html.
49. Young, L.-S. [2003] "Entropy in dynamical systems, " Entropy, Princeton Ser. Appl. Math. (Princeton Univ. Press, Princeton, NJ), pp. 313-327.