Article contents
Escaping orbits are rare in the quasi-periodic Fermi–Ulam ping-pong
Part of:
Nonlinear dynamics
Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Ergodic theory
Hamiltonian and Lagrangian mechanics
Low-dimensional dynamical systems
Published online by Cambridge University Press: 06 September 2018
Abstract
We consider the quasi-periodic Fermi–Ulam ping-pong model with no diophantine condition on the frequencies and show that typically the set of initial data which leads to escaping orbits has Lebesgue measure zero.
MSC classification
- Type
- Original Article
- Information
- Copyright
- © Cambridge University Press, 2018
References
Campos, J. and Tarallo, M.. Nonmonotone equations with large almost periodic forcing terms. J. Differential Equations 254 (2013), 686–724.Google Scholar
De Simoi, J.. Stability and instability results in a model of Fermi acceleration. Discrete Contin. Dyn. Syst. 25 (2009), 719–750.Google Scholar
De Simoi, J.. Fermi acceleration in anti-integrable limits of the standard map. Comm. Math. Phys. 321 (2013), 703–745.Google Scholar
Dolgopyat, D.. Bouncing balls in non-linear potentials. Discrete Contin. Dyn. Syst. 22 (2008), 165–182.Google Scholar
Dolgopyat, D.. Fermi acceleration. Geometric and Probabilistic Structures in Dynamics. Eds. Burns, K., Dolgopyat, D. and Pesin, Y.. American Mathematical Society, Providence, RI, 2008, pp. 149–166.Google Scholar
Dolgopyat, D.. Lectures on Bouncing Balls, lecture notes for a course in Murcia, 2013; available athttp://www2.math.umd.edu/∼dolgop/BBNotes.pdf.Google Scholar
Dolgopyat, D. and De Simoi, J.. Dynamics of some piecewise smooth Fermi–Ulam models. Chaos 22 (2012), 026124.Google Scholar
Einsiedler, M and Ward, T.. Ergodic Theory With a View Towards Number Theory. Springer, Berlin–New York, 2011.Google Scholar
Kunze, M. and Ortega, R.. Complete orbits for twist maps on the plane: the case of small twist. Ergod. Th. & Dynam. Sys. 31 (2011), 1471–1498.Google Scholar
Laederich, S. and Levi, M.. Invariant curves and time-dependent potentials. Ergod. Th. & Dynam. Sys. 11 (1991), 365–378.Google Scholar
Moser, J.. On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1962), 1–20.Google Scholar
Oxtoby, J. C. and Ulam, S. M.. Measure-preserving homeomorphisms and metrical transitivity. Ann. of Math. (2) 42 (1941), 874–920.Google Scholar
Pustylnikov, L. D.. Poincaré models, rigorous justification of the second law of thermodynamics from mechanics, and the Fermi acceleration mechanism. Russian Math. Surveys 50 (1995), 145–189.Google Scholar
Ulam, S. M.. On some statistical properties of dynamical systems. Proc. Fourth Berkeley Symp. on Mathematical Statistics and Probability (Contributions to Astronomy, Meteorology, and Physics, 3). University of California Press, Berkeley, CA, 1961, pp. 315–320.Google Scholar
Zharnitsky, V.. Instability in Fermi–Ulam ‘ping-pong’ problem. Nonlinearity 11 (1998), 1481–1487.Google Scholar
Zharnitsky, V.. Invariant curve theorem for quasiperiodic twist mappings and stability of motion in the Fermi–Ulam problem. Nonlinearity 13 (2000), 1123–1136.Google Scholar
- 8
- Cited by