Abstract
Dynamical systems with fractal geometry can be constructed in a variety of ways: We illustrate this variety with examples based on the Cantor set, the Sierpinski gasket, and on lattices of these fractal-based structures. Depending on the physical parameters, the models can exhibit both discrete and continuous spectra.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Y. Gefen, B. B. Mandelbrot, and A. Aharony,Phys. Rev. Lett. 45:855 (1980).
T. Nakayama, K. Yakubo, and R. L. Orbach,Rev. Mod. Phys. 66:381 (1994).
M. Kohmoto, L. P. Kadanoff, and C. Tang,Phys. Rev. Lett. 50:1870 (1983).
S. Ostlund, R. Pandit, D. Rand, H. J. Schellnhuber, and E. D. Siggia,Phys. Rev. Lett. 50:1873(1983).
B. Sutherland and M. Kohmoto,Phys. Rev. B36:5877 (1987).
M. Moulopoulos and S. Roch,Phys. Rev. B53:212 (1996).
G. Gumbs, G. S. Dubey, A. Salmon, B. S. Mahmoud, and D. Huang,Phys. Rev. B52:210 (1995).
H. Hiramoto and M. Kohmoto,Int. J. Mod. Phys. B6:281 (1992).
Y. Hu, D.-C. Tian, and L. Wang,Phys. Lett. A207:293 (1995).
F. Piechon,Phys. Rev. Lett. 76: 4371 (1996).
A. Hof, O. Knill, and B. Simon,Commun. Math. Phys. 174:149 (1995).
W. P. Keirstead, H. A. Ceccato, and B. A. Huberman,J. Stat. Phys. 53:733 (1988).
P. Kappertz, R. F. S. Andrade, and H. J. Schellnhuber,Phys. Rev. B49:14711 (1994).
F. Cracium et al.,Phys Rev. Lett. 68:1444 (1992);Phys. Rev. B49:15067 (1994).
R. Rammal,Journal de Physique 45:191 (1984).
E. Domany, S. Alexander, D. Benisimon, and L. P. Kadanoff,Phys. Rev. B28:3110 (1983).
B. W. Southern and A. R. Douchout,Phys. Rev. Lett. 55:966 (1985).
K. Yakubo,Phys. Rev. B42:1078 (1990).
X. R. Wang,Phys. Rev. B51:9310 (1995).
D. J. Bergman and Y. Kantor,Phys. Rev. Lett. 53:511 (1984).
P. J.-M. Monceau and J.-C. S. Levy,Phys. Rev. B49:1026 (1994).
W. A. Schwalm, C. C. Reese, C. J. Wagner, and M. K. Schwalm,Phys. Rev. B49:15650 (1994).
C. S. Jayanthi and S. Y. Wu,Phys. Rev. B50:897 (1994).
J. Q. You, C.-H. Lam, F. Nori, and L. M. Sander,Phys. Rev. E48: 4183 (1993).
F. S. de Menezes and A. C. N. de Magalhaes,Phys. Rev. B46:11642 (1992); see also the extensive references in this paper.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kimball, J.C., Frisch, H.L. Vibrations of simple fractal-based models. J Stat Phys 89, 453–468 (1997). https://doi.org/10.1007/BF02770776
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02770776