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
About the Author:
Ivan I. Shevchenko is Professor at Saint Petersburg State University and Head of the Department of Celestial Mechanics and Dynamical Astronomy at Pulkovo Observatory of the Russian Academy of Sciences, Saint Petersburg, Russia. He is author of the book "The Lidov-Kozai Effect - Applications in Exoplanet Research and Dynamical Astronomy", Springer (2017).
