Dynamical Chaos in Planetary Systems

This is the first monograph dedicated entirely to problems of stability and chaotic behaviour in planetary systems and its subsystems. The author explores the three rapidly developing interplaying fields of resonant and chaotic dynamics of Hamiltonian systems, the dynamics of Solar system bodies, and the dynamics of exoplanetary systems. The necessary concepts, methods and tools used to study dynamical chaos (such as symplectic maps, Lyapunov exponents and timescales, chaotic diffusion rates, stability diagrams and charts) are described and then used to show in detail how the observed dynamical architectures arise in the Solar system (and its subsystems) and in exoplanetary systems. The book concentrates, in particular, on chaotic diffusion and clearing effects. The potential readership of this book includes scientists and students working in astrophysics, planetary science, celestial mechanics, and nonlinear dynamics.

Autorentext

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).

Inhalt
Preface 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 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</p&...