Lyapunov exponents and subspace evolution
Authors:
M. Rokni and B. S. Berger
Journal:
Quart. Appl. Math. 45 (1987), 789-793
MSC:
Primary 34D05; Secondary 34C35, 58F10
DOI:
https://doi.org/10.1090/qam/917027
MathSciNet review:
917027
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Differential equations are derived which describe the evolution of area tensors and normals associated with the subspaces of an $n$-dimensional Euclidean phase space, ${E_n}$. These provide computational methods for determining the Lyapunov exponents of continuous dynamical systems.
- Jack K. Hale, Ordinary differential equations, 2nd ed., Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980. MR 587488
- J. L. Synge and A. Schild, Tensor Calculus, Mathematical Expositions, no. 5, University of Toronto Press, Toronto, Ont., 1949. MR 0033165
E. Cartan, Leçons sur la géométrie des espaces de Riemann, Gauthier-Villars, Paris, 1963, Geometry of Riemannian spaces, Math. Sci. Press, 1983
- C. Truesdell and R. Toupin, The classical field theories, Handbuch der Physik, Bd. III/1, Springer, Berlin, 1960, pp. 226–793; appendix, pp. 794–858. With an appendix on tensor fields by J. L. Ericksen. MR 0118005
- A. Cemal Eringen (ed.), Continuum physics. Vol. I, Academic Press, New York-London, 1971. Mathematics. MR 0468443
- David Lovelock and Hanno Rund, Tensor, differential forms, and variational principles, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1975. Pure and Applied Mathematics. MR 0474046
G. Benettin, L. Galagani, A. Giorgilli, J. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; method for computing all of them, Meccanica 15, 9–20 (1980)
- V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210 (Russian). MR 0240280
- B. S. Berger and M. Rokni, Lyapunov exponents and continuum kinematics, Internat. J. Engrg. Sci. 25 (1987), no. 10, 1251–1257. MR 912603, DOI https://doi.org/10.1016/0020-7225%2887%2990045-0
E. N. Lorenz, Deterministic nonperiodic flow, J. Atm. Sci. 20, 130–141 (1963)
O. E. Rossler, An equation for continuous chaos, Phys. Lett. 57A, 397–398 (1976)
- O. E. Rössler, An equation for hyperchaos, Phys. Lett. A 71 (1979), no. 2-3, 155–157. MR 588951, DOI https://doi.org/10.1016/0375-9601%2879%2990150-6
A. Wolf, B. Swift, H. L. Swinney, J. A. Vastand, Determining Lyapunov exponents from a time series, Physica 16D, 285–317 (1985)
- Ippei Shimada and Tomomasa Nagashima, A numerical approach to ergodic problem of dissipative dynamical systems, Progr. Theoret. Phys. 61 (1979), no. 6, 1605–1616. MR 539440, DOI https://doi.org/10.1143/PTP.61.1605
J. Hale, Ordinary differential equations, Krieger Pub. Co., 1982
J. L. Synge and A. Schild, Tensor calculus, University of Toronto Press, 1962
E. Cartan, Leçons sur la géométrie des espaces de Riemann, Gauthier-Villars, Paris, 1963, Geometry of Riemannian spaces, Math. Sci. Press, 1983
C. Truesdell and R. A. Toupin, Classical field theories, Springer, 1960
A. C. Eringen, Continuum physics, Vol. II, Academic Press, 1975
D. Lovelock and H. Rund, Tensors, differential forms and variational principles, J. Wiley, New York, 1975
G. Benettin, L. Galagani, A. Giorgilli, J. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; method for computing all of them, Meccanica 15, 9–20 (1980)
V. I. Oseledec, A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19, 197–231 (1968)
B. S. Berger and M. Rokni, Lyapunov exponents and continuum kinematics, Int. J. Eng. Sci. 25, 1079–1084 (1987)
E. N. Lorenz, Deterministic nonperiodic flow, J. Atm. Sci. 20, 130–141 (1963)
O. E. Rossler, An equation for continuous chaos, Phys. Lett. 57A, 397–398 (1976)
O. E. Rossler, An equation for hyperchaos, Phys. Lett. 71A, 155–157 (1979)
A. Wolf, B. Swift, H. L. Swinney, J. A. Vastand, Determining Lyapunov exponents from a time series, Physica 16D, 285–317 (1985)
I. Shimada, T. Nagashima, A numerical approach to the ergodic problem of dissipative dynamical systems, Prog. in Theoretical Physics 61, 6, 1605–1616 (1979)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
34D05,
34C35,
58F10
Retrieve articles in all journals
with MSC:
34D05,
34C35,
58F10
Additional Information
Article copyright:
© Copyright 1987
American Mathematical Society
