Dynamical Chaos in Planetary Systems

I 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

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