This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The book is intended as a self consistent treatment of the subject at the graduate level and as a reference for scientists already working in the field. It emphasizes both methods of calculation and results. It is accessible to physicists and engineers without training in modern mathematics. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects. It can be used as a graduate text for a two semester course covering both Hamiltonian and dissipative dynamics.

Regular and Chaotic Dynamics

1. Introduction 2. Radial velocities 3. Astrometry 4. Timing 5. Microlensing 6. Transits 7. Imaging 8. Host stars 9. Brown dwarfs and free-floating planets 10. Formation and evolution 11. Interiors and atmospheres 12. The Solar System Appendixes References Index.

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The Exoplanet Handbook

Review and comparison of active space debris capturing and removal methods

Is there chaos in the brain? II. Experimental evidence and related models.

Fundamental theory: Holomorphic functions and domains of holomorphy Implicit functions and analytic sets The Poincare, Cousin, and Runge problems Pseudoconvex domains and pseudoconcave sets Holomorphic mappings Theory of analytic spaces: Ramified domains Analytic sets and holomorphic functions Analytic spaces Normal pseudoconvex spaces Bibliography Index.

Function theory in several complex variables

The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping

This book presents classical celestial mechanics and its interplay with dynamical systems in a way suitable for advance level undergraduate students as well as postgraduate students and researchers First paradigmatic models are used to introduce the reader to the concepts of order, chaos, invariant curves, cantori Next the main numerical methods to investigate a dynamical system are presented Then the author reviews the classical two-body problem and proceeds to explore the three-body model in order to investigate orbital resonances and Lagrange solutions In rotational dynamics the author details the derivation of the rigid body motion, and continues by discussing related topics, from spin-orbit resonances to dumbbell satellite dynamics
 
Perturbation theory is then explored in full detail including practical examples of its application to finding periodic orbits, computation of the libration in longitude of the Moon The main ideas of KAM theory are provided including a presentation of long-term stability and converse KAM results Celletti then explains the implementation of computer-assisted techniques, which allow the user to obtain rigorous results in good agreement with the astronomical expectations Finally the study of collisions in the solar system is approached

Stability and chaos in celestial mechanics

KAM theory is a powerful tool apt to prove perpetual stability in Hamiltonian systems, which are a perturbation of integrable ones The smallness requirements for its applicability are well known to be extremely stringent A long standing problem, in this context, is the application of KAM theory to "physical systems" for "observable" values of the perturbation parameters
We consider the Restricted, Circular, Planar, Three-Body Problem (RCP3BP), ie, the problem of studying the planar motions of a small body subject to the gravitational attraction of two primary bodies revolving on circular Keplerian orbits (which are assumed not to be influenced by the small body) When the mass ratio of the two primary bodies is small, the RCP3BP is described by a nearly-integrable Hamiltonian system with two degrees of freedom; in a region of phase space corresponding to nearly elliptical motions with non small eccentricities, the system is well described by Delaunay variables The Sun-Jupiter observed motion is nearly circular and an asteroid of the Asteroidal belt may be assumed not to influence the Sun-Jupiter motion The Jupiter-Sun mass ratio is slightly less than 1/1000
We consider the motion of the asteroid 12 Victoria taking into account only the Sun-Jupiter gravitational attraction regarding such a system as a prototype of a RCP3BP For values of mass ratios up to 1/1000, we prove the existence of two-dimensional KAM tori on a fixed three-dimensional energy level corresponding to the observed energy of the Sun-Jupiter-Victoria system Such tori trap the evolution of phase points "close" to the observed physical data of the Sun-Jupiter-Victoria system As a consequence, in the RCP3BP description, the motion of Victoria is proven to be forever close to an elliptical motion
The proof is based on: 1) a new iso-energetic KAM theory; 2) an algorithm for computing iso-energetic, approximate Lindstedt series; 3) a computer-aided application of 1)+2) to the Sun-Jupiter-Victoria system
The paper is self-contained but does not include the (similar to 12000 lines) computer programs, which may be obtained by sending an e-mail to one of the authors

KAM stability and celestial mechanics

We study the stability of spin-orbit resonances in Celestial Mechanics, namely the exact commensurabilities between the periods of rotation and revolution of satellites or planets We introduce a mathematical model describing an approximation of the physical situation and we select a set of satellites for which such simplified model provides a good approximation

Alessandra Celletti

Papers

The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping

Stability and chaos in celestial mechanics

KAM stability and celestial mechanics

Analysis of resonances in the spin-orbit problem in Celestial Mechanics: the synchronous resonance (part I)

A KAM theory for conformally symplectic systems: Efficient algorithms and their validation