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**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**

**References**

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