Abstract
The method of cell mappings has been developed as an efficient tool for the global study of dynamical systems. One of them, the generalized cell mapping (GCM), describes the behavior of a system in a probabilistic sense, and is essentially a Markov chain analysis of dynamical systems. Since the largest Lyapunov exponent is widely used to characterize attractors of dynamical systems, we propose an algorithm for that quantity by the GCM. This allows us to examine the persistent groups of the GCM in terms of their Lyapunov exponent, thereby connecting them with their counterparts in point mapping systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, 1983).
A. J. Lichtenberg and L. A. Lieberman,Regular and Stochastic Motion (Springer, New York, 1983).
C. S. Hsu,J. Appl. Mech. 47:931 (1980).
C. S. Hsu and R. S. Guttalu,J. Appl. Mech. 47:940 (1980).
C. S. Hsu,J. Appl. Mech. 48:634 (1981).
C. S. Hsu, R. S. Guttalu, and W. H. Zhu,J. Appl. Mech. 49:885 (1982).
C. S. Hsu and M. C. Kim,J. Stat. Phys. 38:735 (1985).
H. M. Chiu,Random vibration analysis by cell mapping methods, Ph. D. thesis, Department of Mechanical Engineering, University of California (1984).
C. S. Hsu and M. C. Kim,Phys. Rev. A 30:3351 (1984).
C. S. Hsu and M. C. Kim,Phys. Rev. A 31:3253 (1985).
J. E. Marsden and M. McCracken,The Hopf Bifurcation and Its Applications (Springer, New York, 1976).
V. I. Oseledec,Trans. Moscow Math. Soc. 19:197 (1968).
J. P. Eckmann and D. Ruelle,Rev. Mod. Phys. 57:617 (1985).
G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn,Meccanica 15:21 (1980).
R. Abraham, J. E. Marsden, and T. Ratiu,Manifolds, Tensor Analysis, and Applications (Addison-Wesley, Reading, Massachusetts, 1983).
E. Ott,Rev. Mod. Phys. 53:655 (1981).
M. Hénon,Commun. Math. Phys. 50:69 (1976).
Y. Ueda,New Approaches to Nonlinear Problems in Dynamics, P. Holmes, ed. (SIAM, Philadelphia, 1980), p. 311.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kim, M.C., Hsu, C.S. Computation of the largest Lyapunov exponent by the generalized cell mapping. J Stat Phys 45, 49–61 (1986). https://doi.org/10.1007/BF01033076
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01033076