A system of one dimensional balls with gravity
Article
Received:
Revised:
- 83 Downloads
- 36 Citations
Abstract
We introduce a Hamiltonian system with many degrees of freedom for which the nonvanishing of (some) Lyapunov exponents almost everywhere can be established analytically.
Keywords
Neural Network Statistical Physic Complex System Nonlinear Dynamics Hamiltonian System
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Preview
Unable to display preview. Download preview PDF.
References
- [A-S] Anosov, D. V., Sinai, Ya. G.: Certain smooth ergodic systems. Russ. Math. Surv.22, 103–167 (1967)Google Scholar
- [B-B] Ballmann, W., Brin, M.: On the ergodicity of geodesis flows. Erg. Th. Dyn. Syst.2, 311–315 (1982)Google Scholar
- [B1] Bunimovich, L. A.: On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys.65, 295–312 (1979)CrossRefGoogle Scholar
- [B2] Bunimovich, L. A.: Many-dimensional nowhere dispersing billiards with chaotic behavior. Physica D33, 58–64 (1988)Google Scholar
- [Bu] Burns, K.: Hyperbolic behavior of geodesic flows on manifolds with no focal points. Erg. Th. Dyn. Syst. 3, 1–12 (1983)Google Scholar
- [B-E] Burton, R., Easton, R. W.: Ergodic properties of linked twist mappings. Lecture Notes in Mathematics, vol.819, pp. 35–49. Berlin, Heidelberg, New York: Springer 1980Google Scholar
- [B-G] Burns, K., Gerber, M. G.: Real analytic Bernoulli geodesic flows onS 2. Erg. Th. Dyn. Syst. (to appear)Google Scholar
- [Ch-S] Chernov, N. I., Sinai, Ya. G.: Entropy of the gas of hard spheres with respect to the group of time space translations. Proceedings of the I. G. Petrovsky Seminar Vol.8, 218–238 (1982)Google Scholar
- [C-F-S] Cornfeld, I. P., Fomin, S. V., Sinai, Ya. G.: Ergodic Theory. Berlin, Heidelberg, New York: Springer 1982Google Scholar
- [De1] Devaney, R. L.: A piecewise linear model for the zones of instability of an area presenting map. PhysicaD10, 383–393 (1984)Google Scholar
- [De2] Devaney, R. L.: Linked twist mappings are almost Anosov. Lecture Notes in Mathematics, vol.819, pp 35–49. Berlin, Heidelberg, New York: Springer 1980Google Scholar
- [Do1] Donnay, V. J.: Convex billiards with positive entropy. (In preparation)Google Scholar
- [Do2] Donnay, V. J.: Geodesic flow on the two-sphere. Part I: Positive measure entropy. Erg. Th. Dyn. Syst.8, 531–553 (1988)Google Scholar
- [D-L] Donnay, V. J., Liverani, C.: Ergodic properties of particle motion in potential fields. (Preprint 1989)Google Scholar
- [G] Gerber, M.: Conditional stability and real analytic pseudo-Anosov maps. AMS Memoirs, No.321, (1985)Google Scholar
- [H-D-M-S] Hayli, A., Dumont, T., Moulin-Ollagnier, J., Strelcyn, J.-M.: Quelques resultats nouveaux sur les billiards de Robnik. J. Phys. A: Math. Gen.20, 3237–3249 (1987)CrossRefGoogle Scholar
- [Kn] Knauf, A.: Ergodic and topological properties of Coulombic periodic potentials. Commun. Math. Phys.110, 89–112 (1987)CrossRefGoogle Scholar
- [K-S] Katok, A. Strelcyn, J.-M., with collaboration of Ledrappier, F., Przytycki, F.: Invariant manifolds, entropy and billiards; smooth maps with singularities. Lecture Notes in Mathematics, vol1222. Berlin, Heidelberg, New York: Springer 1986Google Scholar
- [L-M] Lehtihet, H. E., Miller, B. N.: Numerical study of a billiard in a gravitational field. PhysicaD21, 94–104 (1986)Google Scholar
- [M] Markarian, R.: Billiards with Pesin region of measure one. Commun. Math. Phys.118, 87–97 (1988)CrossRefGoogle Scholar
- [O] Oseledets, V. I.: The multiplicative ergodic theorem. The Lyapunov characteristic numbers of a dynamical system. Trans. Mosc. Math. Soc.19, 197–231 (1968)Google Scholar
- [P] Pesin, Ya. B.: Lyapunov characteristic exponents and smooth ergodic theory. Russ. Math. Surv.32, 55–114 (1977)Google Scholar
- [Pr1] Przytycki, F.: Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviors. Ergod. Th. Dyn. Syst.2, 439–463 (1982)Google Scholar
- [Pr2] Przytycki, F.: Ergodicity of toral linked twist mappings. Ann. Sci. Ecole Norm Sup.16, 345–354 (1983)Google Scholar
- [Ro] Robnik, M.: Classical dynamics of a family of billiards with analytic boundaries. J. Phys. A,16, 3971–3986 (1983)Google Scholar
- [Ru] Ruelle, D.: Ergodic theory of differentiable dynamical systems. Publ. Math. IHES50, 27–58 (1979)Google Scholar
- [S1] Sinai, Ya. G.: Development of Krylov's ideas. Afterword to N. S. Krylov, Works on the foundations of statistical physics. Princeton, NJ: Princeton University Press 1979Google Scholar
- [S2] Sinai, Ya. G.: Dynamical Systems with elastic reflections. Ergodic properties of dispersing billiards. Russ. Math. Surv.25, 137–189 (1970)Google Scholar
- [W1] Wojtkowski, M. P.: A model problem with the coexistence of stochastic and integrable behavior. Commun. Math. Phys.80, 453–464 (1981)CrossRefGoogle Scholar
- [W2] Wojtkowski, M. P.: On the ergodic properties of piecewise linear perturbations of the twist map. Ergod. Th. Dyn. Syst.2, 525–542 (1982)Google Scholar
- [W3] Wojtkowski, M. P.: Measure theoretic entropy of the system of hard spheres. Ergod. Th. Dyn. Syst.8, 133–153 (1988)Google Scholar
- [W4] Wojtkowski, M. P.: Principles for the design of billiards with nonvanishing Lyapunov exponents. Commun. Math. Phys.105, 319–414 (1986)CrossRefGoogle Scholar
- [W5] Wojtkowski, M. P.: Linked twist mappings have theK-property. Annals of the New York Academy of Sciences Vol.357, Nonlinear Dynamics, 65–76 (1980)Google Scholar
- [W6] Wojtkowski, M. P.: Linearly stable orbits in 3 dimensional billiards. (Preprint 1989)Google Scholar
- [W7] Wojtkowski, M. P.: Invariant families of cones and Lyapunov exponents. Ergod. Th. Dyn. Syst.5, 145–161 (1985)Google Scholar
Copyright information
© Springer-Verlag 1990