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Frequency Map Analysis, theory and practice

Jacques Laskar

Astronomie et Systèmes Dynamiques, IMC-CNRS UMR8028,
77 Av. Denfert-Rochereau, 75014 Paris, France

Introduction. Frequency Map Analysis is a numerical method based on refined Fourier techniques which provides a clear representation of the global dynamics of many multi-dimensional systems, and which is particularly adapted for systems of 3-degrees of freedom and more. This method relies heavily on the possibility of making accurate quasiperiodic approximations of quasiperiodic signal given in a numerical way.

In these lectures, we will describe the basis of the frequency analysis method. This is intended to be of practical use for any researcher who is willing to explore and use the frequency map analysis methods for the understanding of the dynamics of Hamiltonian systems.

Applications to several examples will be provided, in Solar System, or Particle Accelerator Dynamics.

1. Quasi periodic approximation

2. Convergence of the quasi periodic approximation, asymptotic expansion

3. Numerical examples

4. Frequency Map Analysis

5. Application of Frequency Map Analysis in Solar System and Particle Accelerator dynamics

Laskar, J. 1990 The chaotic motion of the solar system. A numerical estimate of the size of the chaotic zones Icarus 88 266-291
Dumas, S., Laskar, J. 1993 Global Dynamics and Long-Time Stability in Hamiltonian Systems via Numerical Frequency Analysis Phys. Rev. Lett. 70 2975-2979
Laskar, J. 1993 Frequency analysis for multi-dimensional systems. Global dynamics and diffusion Physica D 67 257-281
Laskar, J. 1999 Introduction to frequency map analysis in proc. of NATO ASI Hamiltonian Systems with Three or More Degrees of Freedom, C. Simò ed, Kluwer 134-150
Laskar, J., Froeschlé, C., Celletti, A. 1992 The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping Physica D 56 253-269
Laskar, J., Robutel, P. 1993 The chaotic obliquity of the planets Nature 361 608-612
Laskar, J., Robin, D. 1996 Application of frequency map analysis to the ALS Particle Accelerator 54 183-192
Papaphilippou, Y., Laskar, J. 1996 Frequency map analysis and global dynamics in a two degrees of freedom galactic potential Astron. Astrophys. 307 427-449
Papaphilippou, Y., Laskar, J. 1998 Global dynamics of triaxial galactic models through frequency map analysis Astron. Astrophys. 329 451-481
Laskar, J.: 2000, Application of frequency map analysis in galactic dynamics, in proc. The Chaotic Universe, Rome, Pescara, 1-5 fev 1999, V. Gurdzadyan and R. Ruffini, eds., World Scientific, 115-126
Robin, D., Steir, C., Laskar, J., Nadolski, L.: 2000, Global dynamics of the ALS revealed through experimental Frequency Map Analysis, Phys. Rev. Let. 85 558-561
Chandre, C., Laskar, J., Benfatto, G., Jauslin, H.R.: 2001, Determination of the breakup of invariant tori in three frequency Hamiltonian systems Physica D 154 159-170
Robutel, P., Laskar, J.: 2001, Frequency Map and Global Dynamics in the Solar System I: Short period dynamics of massless particles, Icarus 152 4-28

next up previous
Next: Matsushita Up: Abstracts Previous: Kuramoto