The Fermi-Pasta-Ulam Problem pp 239-282 | Cite as
Numerical Methods and Results in the FPU Problem
Chapter
Keywords
Lyapunov Exponent Thermodynamic Limit Regular Orbit Trapping Time Toda Chain
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
- 1.D. Bambusi and A. Ponno, On meta-stability in FPU, Comm. Math. Phys, 264, 539–561 (2006).CrossRefADSMathSciNetMATHGoogle Scholar
- 2.G. Benettin, Time scale for energy equipartition in a two-dimensional FPU model, Chaos, 15, 015108, 10 (2005).MathSciNetGoogle Scholar
- 3.G. Benettin and F. Fassó, From Hamiltonian perturbation theory to symplectic integrators and back, Proceedings of the NSF/CBMS Regional Conference on numerical analysis of Hamiltonian differential equations (Golden, CO, 1997), 29, 73–87 (1999).Google Scholar
- 4.G. Benettin, L. Galgani, A. Giorgilli and J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them, part 1: theory, Meccanica, 9–20 (1980).Google Scholar
- 5.G. Benettin, L. Galgani, A. Giorgilli and J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them, part 2: numerical applications, Meccanica, 21–30 (1980).Google Scholar
- 6.G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near-to-the-identity symplectic mappings with application to symplectic integration algorithms, J. Statist. Phys., 74, 1117–1143 (1994).MATHCrossRefMathSciNetADSGoogle Scholar
- 7.L. Berchialla, L. Galgani and A. Giorgilli, Localization of energy in FPU chains,Discrete Contin. Dyn. Syst., 11, 855–866 (2004).MATHMathSciNetCrossRefGoogle Scholar
- 8.L. Berchialla, A. Giorgilli and S. Paleari, Exponentially long times to equipartition in the thermodynamic limit, Phys. Lett. A, 321, 167–172 (2004).MATHCrossRefADSGoogle Scholar
- 9.P. Bocchieri, A. Scotti, B. Bearzi and A. Loinger, Anharmonic chain with Lennard-Jones interaction, Phys. Rev. A, 2, 2013–2019 (1970).CrossRefADSGoogle Scholar
- 10.L.A. Bunimovich, S.G. Dani, R.L. Dobrushin, M.V. Jakobson, I.P. Kornfeld, N.B. Maslova, Y.B. Pesin, Y.G. Sinai, J. Smillie, Y.M. Sukhov and A.M.Vershik, Dynamical systems, ergodic theory and applications, vol. 100 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, revised edition, 2000. Edited and with a preface by Sinai, Translated from the Russian, Mathematical Physics, I.Google Scholar
- 11.A. Carati, L. Galgani, A. Ponno and A. Giorgilli, The Fermi-Pasta-Ulam problem, Nuovo Cimento B (11), 117, 1017–1026 (2002).ADSGoogle Scholar
- 12.L. Casetti, M. Cerruti-Sola, M. Pettini and E.G.D. Cohen, The Fermi–Pasta–Ulam problem revisited: stochasticity thresholds in nonlinear Hamiltonian systems, Phys. Rev. E (3), 55, 6566–6574 (1997).CrossRefADSMathSciNetGoogle Scholar
- 13.C. Cercignani, L. Galgani and A. Scotti, Phys. Lett. A, 38, 403 (1972).CrossRefADSGoogle Scholar
- 14.P. Cincotta, C. Giordano and C. Simó, Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits, Phys. D, 182, 151–178 (2003).MATHCrossRefMathSciNetGoogle Scholar
- 15.J. De Luca, A. Lichtenberg and M.A. Lieberman, Time scale to ergodicity in the fermi-pasta-ulam system, Chaos 5, 283–297 (1995).CrossRefADSGoogle Scholar
- 16.J. De Luca, A. Lichtenberg and S. Ruffo, Energy transition and time scale to equipartition in the Fermi–Pasta–Ulam oscillator chain, Phys. Rev. E (3), 51, 2877–2884 (1995).CrossRefGoogle Scholar
- 17.J. De Luca, A. Lichtenberg and S. Ruffo, Finite times to equipartition in the thermodynamic limit, Phys. Rev. E (3), 60, 3781–3786 (1999).CrossRefGoogle Scholar
- 18.E. Fermi, J. Pasta and S. Ulam, Studies of nonlinear problems, in Collected papers (Notes and memories). Vol. II: United States, 1939–1954, 1955. Los Alamos document LA-1940.Google Scholar
- 19.M. Fouchard, E. Lega, C. Froeschlé and C. Froeschlé, On the relationship between fast Lyapunov indicator and periodic orbits for continuous flows, Celestial Mech. Dynam. Astronom., 83, 205–222, (2002). Modern celestial mechanics: from theory to applications (Rome, 2001).MATHCrossRefADSMathSciNetGoogle Scholar
- 20.C. Froeschlé, M. Guzzo and E. Lega, Graphical evolution of the Arnold web: from order to chaos, Science, 289, 2108–2110 (2000).CrossRefADSGoogle Scholar
- 21.C. Froeschlé and E. Lega, On the structure of symplectic mappings. The fast Lyapunov indicator: a very sensitive tool, Celestial Mech. Dynam. Astronom., 78 (2000), 167–195 (2001). New developments in the dynamics of planetary systems (Badhofgastein, 2000).ADSGoogle Scholar
- 22.C. Froeschlé, E. Lega and R. Gonczi, Fast Lyapunov indicators. Application to asteroidal motion, Celestial Mech. Dynam. Astronom., 67, 41–62 (1997).MATHCrossRefADSMathSciNetGoogle Scholar
- 23.F. Fucito, F. Marchesoni, E. Marinari, G. Parisi, L. Peliti, S. Ruffo and A. Vulpiani, Approach to equilibrium in a chain of nonlinear oscillators, J. Physique, 43, 707–713 (1982).CrossRefMathSciNetGoogle Scholar
- 24.L. Galgani, A. Giorgilli, A. Martinoli and S. Vanzini, On the problem of energy equipartition for large systems of the Fermi-Pasta-Ulam type: analytical and numerical estimates, Phys. D, 59, 334–348 (1992).MATHCrossRefMathSciNetGoogle Scholar
- 25.L. Galgani and A. Scotti, Phys. Rev. Lett., 28, 1173 (1972).CrossRefADSGoogle Scholar
- 26.L. Galgani and A. Scotti, Recent progress in classical nonlinear dynamics, Riv. Nuovo Cimento (2), 2, 189–209 (1972).MathSciNetADSCrossRefGoogle Scholar
- 27.A. Giorgilli, S. Paleari and T. Penati, Local chaotic behaviour in the FPU system,Discrete Contin. Dyn. Syst. Ser. B, 5, 991–1004 (2005).MATHMathSciNetCrossRefGoogle Scholar
- 28.M. Guzzo, E. Lega and C. Froeschlé, On the numerical detection of the effective stability of chaotic motions in quasi-integrable systems, Phys. D, 163, 1–25 (2002).MATHCrossRefMathSciNetGoogle Scholar
- 29.M. Guzzo, E. Lega and C. Froeschlé, First numerical evidence of global Arnold diffusion in quasi-integrable systems, Discrete Contin. Dyn. Syst. Ser. B, 5, 687–698 (2005).MATHCrossRefGoogle Scholar
- 30.E. Hairer, Backward analysis of numerical integrators and symplectic methods, Ann. Numer. Math., 1, 107–132 (1994). Scientific computation and differential equations (Auckland, 1993).MATHMathSciNetGoogle Scholar
- 31.E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration, vol. 31 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2002. Structure-preserving algorithms for ordinary differential equations.Google Scholar
- 32.F.M. Izrailev and B.V. Chirikov, Stochasticity of the simplest dynamical model with divided phase space, Sov. Phys. Dokl., 11, 30 (1966).ADSGoogle Scholar
- 33.H. Kantz, Vanishing stability thresholds in the thermodynamic limit of nonintegrable conservative systems, Phys. D, 39, 322–335 (1989).CrossRefMathSciNetMATHGoogle Scholar
- 34.H. Kantz, R. Livi and S. Ruffo, Equipartition thresholds in chains of anharmonic oscillators, J. Statis. Phys., 76, 627–643 (1994).CrossRefMathSciNetADSGoogle Scholar
- 35.E. Lega, M. Guzzo and C. Froeschlé, Detection of Arnold diffusion in Hamiltonian systems, Phys. D, 182, 179–187 (2003).MATHCrossRefMathSciNetGoogle Scholar
- 36.V.I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obv sv c., 19, 179–210 (1968).MathSciNetGoogle Scholar
- 37.S. Paleari and T. Penati, Relaxation time to equilibrium in Fermi–Pasta–Ulam system, in Symmetry and perturbation theory (Cala Gonone, 2004), World Science Publishing, River Edge, NJ, 2004, pp. 255–263.Google Scholar
- 38.S. Paleari and T. Penati, Equipartition times in a Fermi–Pasta–Ulam system, Discr. Contin. Dyn. Syst., (2005). Dynamical systems and differential equations (Pomona, CA, 2004).Google Scholar
- 39.M. Pettini and M. Landolfi, Relaxation properties and ergodicity breaking in nonlinear Hamiltonian dynamics, Phys. Rev. A (3), 41, 768–783 (1990).CrossRefADSMathSciNetGoogle Scholar
- 40.B. Rink, Symmetry and resonance in periodic FPU chains, Comm. Math. Phys. 218, 665–685 (2001).MATHCrossRefADSMathSciNetGoogle Scholar
- 41.C. Skokos, Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits, J. Phys. A, 34, 10029–10043 (2001).MATHCrossRefADSMathSciNetGoogle Scholar
- 42.C. Skokos, C. Antonopoulos, T.C. Bountis and M.N. Vrahatis, Detecting order and chaos in Hamiltonian systems by the SALI method, J. Phys. A, 37, 6269–6284 (2004).CrossRefADSMathSciNetGoogle Scholar
- 43.K. Ullmann, A. Lichtenberg and G. Corso, Energy equipartition starting from high-frequency modes in the Fermi–Pasta–Ulam β oscillator chain, Prhys. Rev. E (3), 61, 2471–2477 (2000).CrossRefADSGoogle Scholar
- 44.L. Verlet, Computer “experiments” on classical fluids. I.Thermodynamical properties of Lennard-Jones molecules, Phys. Rev., 159, 98–103 (1967).CrossRefADSGoogle Scholar
Copyright information
© Springer-Verlag Berlin Heidelberg 2007