Smooth dynamics on Weierstrass nowhere differentiable curves
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- by Brian R. Hunt and James A. Yorke PDF
- Trans. Amer. Math. Soc. 325 (1991), 141-154 Request permission
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
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O. Decroly and A. Goldbeter, Multiple periodic regimes and final state sensitivity in a biochemical system, Phys. Lett. A 105 (1984), 196-200.
- Celso Grebogi, Edward Ott, and James A. Yorke, Basin boundary metamorphoses: changes in accessible boundary orbits, Phys. D 24 (1987), no. 1-3, 243–262. MR 887851, DOI 10.1016/0167-2789(87)90078-9
- Celso Grebogi, Edward Ott, and James A. Yorke, Chaotic attractors in crisis, Phys. Rev. Lett. 48 (1982), no. 22, 1507–1510. MR 659056, DOI 10.1103/PhysRevLett.48.1507
- Celso Grebogi, Edward Ott, and James A. Yorke, Crises, sudden changes in chaotic attractors, and transient chaos, Phys. D 7 (1983), no. 1-3, 181–200. Order in chaos (Los Alamos, N.M., 1982). MR 719052, DOI 10.1016/0167-2789(83)90126-4
- Celso Grebogi, Edward Ott, and James A. Yorke, Critical exponent of chaotic transients in nonlinear dynamical systems, Phys. Rev. Lett. 57 (1986), no. 11, 1284–1287. MR 856391, DOI 10.1103/PhysRevLett.57.1284
- Celso Grebogi, Edward Ott, and James A. Yorke, Fractal basin boundaries, long-lived chaotic transients, and unstable-unstable pair bifurcation, Phys. Rev. Lett. 50 (1983), no. 13, 935–938. MR 699395, DOI 10.1103/PhysRevLett.50.935 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.
- G. H. Hardy, Weierstrass’s non-differentiable function, Trans. Amer. Math. Soc. 17 (1916), no. 3, 301–325. MR 1501044, DOI 10.1090/S0002-9947-1916-1501044-1
- R. G. Holt and I. B. Schwartz, Newton’s method as a dynamical system: global convergence and predictability, Phys. Lett. A 105 (1984), no. 7, 327–333. MR 766027, DOI 10.1016/0375-9601(84)90273-1
- James L. Kaplan, John Mallet-Paret, and James A. Yorke, The Lyapunov dimension of a nowhere differentiable attracting torus, Ergodic Theory Dynam. Systems 4 (1984), no. 2, 261–281. MR 766105, DOI 10.1017/S0143385700002431
- Steven W. McDonald, Celso Grebogi, Edward Ott, and James A. Yorke, Fractal basin boundaries, Phys. D 17 (1985), no. 2, 125–153. MR 815280, DOI 10.1016/0167-2789(85)90001-6
- Steven W. McDonald, Celso Grebogi, Edward Ott, and James A. Yorke, Structure and crises of fractal basin boundaries, Phys. Lett. A 107 (1985), no. 2, 51–54. MR 774894, DOI 10.1016/0375-9601(85)90193-8
- 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), no. 14, 1439–1442. MR 804660, DOI 10.1103/PhysRevLett.55.1439
- Ira B. Schwartz, Bistability, basins of attraction and predictability in a forced mass-reaction model, Phys. Lett. A 106 (1984), no. 8, 339–342. MR 776562, DOI 10.1016/0375-9601(84)90912-5 S. Takesue and K. Kaneko, Fractal basin structure, Prog. Theoret. Phys. 71 (1984), 35-49.
- Y. Yamaguchi and N. Mishima, Fractal basin boundary of a two-dimensional cubic map, Phys. Lett. A 109 (1985), no. 5, 196–200. MR 793138, DOI 10.1016/0375-9601(85)90301-9
Additional Information
- © Copyright 1991 American Mathematical Society
- 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