PrefaceI Origins and manifestations of dynamical chaos 11
1 Chaotic behaviour 13
1.1 Pendulum, resonances and chaos 13
1.2 Models of resonance ...... . 15
1.3 Interaction and overlap of resonances . 15
1.4 Symplectic maps in general 16
1.5 The standard map ....... . 18
1.6 The separatrix map ...... . 19
1.7 The separatrix algorithmic map 23
1.8 Geometry of chaotic layers . . . . 26
2 Numerical tools for studies of dynamical chaos 41
2.1 The Lyapunov exponents ....... . 41
2.2 The Poincare sections ......... . 50
2.3 Stability diagrams and dynamical charts 51
2.4 Statistics of Poincare recurrences 51
3 Adiabatic and non-adiabatic chaos: the Lyapunov timescales 53
3.1 Non-adiabatic chaos ... . 54
3.1.1 Chirikov's constant .... . 54
3.2 Adiabatic chaos .......... . 62
3.3 The Lyapunov timescales in resonance doublets and triplets 71
3.4 The Lyapunov exponents in resonance multiplets 74
4 Chaotic diffusion 79
4.1 Diffusion rates 79
4.1.1 Diffusion rates in resonance multiplets ..... . 79
4.1.2 Diffusion rates in resonance triplets and doublets 81
5 Lyapunov and diffusion timescales: relationships 85
5.1 Finite-time Lyapunov exponents 87
5.2 The generic relationship .... 87
5.3 Conditions for the relationship 90
5.4 Numerical examples . . . . . . &n
