Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Smooth dynamics on Weierstrass nowhere differentiable curves


Authors: Brian R. Hunt and James A. Yorke
Journal: Trans. Amer. Math. Soc. 325 (1991), 141-154
MSC: Primary 58F13; Secondary 54H20
DOI: https://doi.org/10.1090/S0002-9947-1991-1040043-X
MathSciNet review: 1040043
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a family of smooth maps on an infinite cylinder which have invariant curves that are nowhere smooth. Most points on such a curve are buried deep within its spiked structure, and the outermost exposed points of the curve constitute an invariant subset which we call the "facade" of the curve. We find that for surprisingly many of the maps in the family, all points in the facades of their invariant curves are eventually periodic.


References [Enhancements On Off] (What's this?)

  • [1] O. Decroly and A. Goldbeter, Multiple periodic regimes and final state sensitivity in a biochemical system, Phys. Lett. A 105 (1984), 196-200.
  • [2] C. Grebogi, E. Ott, and J. A. Yorke, Basin boundary metamorphoses: changes in accessible boundary orbits, Phys. D 24 (1987), 243-262. MR 887851 (88g:58104)
  • [3] -, Chaotic attractors in crisis, Phys. Rev. Lett. 48 (1982), 1507-1510. MR 659056 (83i:58064)
  • [4] -, Crises, sudden changes in chaotic attractors, and transient chaos, Phys. D 7 (1983), 181-200. MR 719052 (85d:58062)
  • [5] -, Critical exponent of chaotic transients in nonlinear dynamical systems, Phys. Rev. Lett. 57 (1986), 1284-1287. MR 856391 (87i:58115)
  • [6] -, Fractal basin boundaries, long-lived chaotic transients, and unstable-unstable pair bifurcation, Phys. Rev. Lett. 50 (1983), 935--938. MR 699395 (84f:58089a)
  • [7] E. G. Gwinn and R. M. Westervelt, Intermittent chaos and low frequency noise in the driven damped pendulum, Phys. Rev. Lett. 54 (1985), 1613-1616.
  • [8] G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc. 17 (1916), 301-325. MR 1501044
  • [9] R. G. Holt and I. B. Schwartz, Newton's method as a dynamical system: global convergence and predictability, Phys. Lett. A 105 (1984), 327-333. MR 766027 (86d:58056)
  • [10] J. L. Kaplan, J. Mallet-Paret, and J. A. Yorke, The Lyapunov dimension of a nowhere differentiable attracting torus, Ergodic Theory Dynamical Systems 4 (1984), 261-281. MR 766105 (86h:58091)
  • [11] S. W. McDonald, C. Grebogi, E. Ott, and J. A. Yorke, Fractal basin boundaries, Phys. D 17 (1985), 125-153. MR 815280 (87k:58170)
  • [12] -, Structure and crises of fractal basin boundaries, Phys. Lett. A 107 (1985), 51-54. MR 774894 (86b:58084)
  • [13] F. C. Moon and G.-X. Li, Fractal basin boundaries and homoclinic orbits for periodic motion in a two-well potential, Phys. Rev. Lett. 55 (1985), 1439-1442. MR 804660 (86m:58096)
  • [14] I. B. Schwartz, Bistability, basins of attraction and predictability in a forced mass-reaction mode, Phys. Lett. A 106 (1984), 339-342. MR 776562 (86d:58101)
  • [15] S. Takesue and K. Kaneko, Fractal basin structure, Prog. Theoret. Phys. 71 (1984), 35-49.
  • [16] Y. Yamaguchi and N. Mishima, Fractal basin boundary of a two-dimensional cubic map, Phys. Lett. A 105 (1984), 259-262. MR 793138 (86j:58106)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F13, 54H20

Retrieve articles in all journals with MSC: 58F13, 54H20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1040043-X
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society