KAM theory for conformally symplectic systems

We present a KAM theory for some dissipative systems (geometrically, these are conformally symplectic systems, i.e. systems that transform a symplectic form into a multiple of itself). For systems with n degrees of freedom depending on n parameters we show that it is possible to find solutions with n-dimensional (Diophantine) frequencies by adjusting the parameters. We do not assume that the system is close to integrable, but we use an a-posteriori format. Our unknowns are a parameterization of the solution and a parameter. We show that if there is a sufficiently approximate solution of the invariance equation, which also satisfies some explicit non–degeneracy conditions, then there is a true solution nearby. We present results both in Sobolev norms and in analytic norms. The a–posteriori format has several consequences: A) smooth dependence on the parameters, including the singular limit of zero dissipation; B) estimates on the measure of parameters covered by quasi–periodic solutions; C) convergence of perturbative expansions in analytic systems; D) bootstrap of regularity (i.e., that all tori which are smooth enough are analytic if the map is analytic); E) a numerically efficient criterion for the break–down of the quasi–periodic solutions. The proof is based on an iterative quadratically convergent method and on suitable estimates on the (analytical and Sobolev) norms of the approximate solution. The iterative step takes advantage of some geometric identities, which give a very useful coordinate system in the neighborhood of invariant (or approximately invariant) tori. This system of coordinates has several other uses: A) it shows that for dissipative conformally symplectic systems the quasi–periodic solutions are attractors, B) it leads to efficient algorithms, which have been implemented elsewhere. Details of the proof are given mainly for maps, but we also explain the slight modifications needed for flows and we devote the appendix to present explicit algorithms for flows.

山陵都監儀軌. 산릉도감의궤. San lūñ do kam ūi kue

Cérémonial prévu par les conseillers pour la construction du mausolée Ui Nung du roi Kyong Jong

경종 의릉 산릉도감의궤景宗 懿陵 山陵都監儀軌San lūñ do kam ūi kue Construction du mausolée Ui Nung du roi Kyong Jong

Premier volume du cérémonial prévu par les conseillers pour la construction du mausolée Ui Nung du roi Kyong Jong.

인선왕후 국장도감의궤. 仁宣王后 國葬都監儀軌. In sōn kuk čañ do kam ūi kue

Troisième volume du cérémonial prévu par les conseillers pour les funérailles nationales de la Reine In Sōn.

인조 장릉 천릉도감도청의궤. 仁祖 長陵遷陵都監都廳儀軌. In čo čañ lūñ čʹōn lūñ do kam do čʹōñ ūi kue

Deuxième des sept volumes du cérémonial prévu par les conseillers pour la translation du mausolée Jan Lung du roi In Jo et de son épouse la reine Jang Lyol.

KAM: Automatic Planning and Interpretation of Numerical Experiments Using Geometrical Methods

KAM is a computer program that can automatically plan, monitor, and interpret numerical experiments with Hamiltonian systems with two degrees of freedom. The program has recently helped solve an open problem in hydrodynamics. Unlike other approaches to qualitative reasoning about physical system dynamics, KAM embodies a significant amount of knowledge about nonlinear dynamics. KAM's ability to control numerical experiments arises from the fact that it not only produces pictures for us to see, but also looks at (sic---in its mind's eye) the pictures it draws to guide its own actions. KAM is organized in three semantic levels: orbit recognition, phase space searching, and parameter space searching. Within each level spatial properties and relationships that are not explicitly represented in the initial representation are extracted by applying three operations ---(1) aggregation, (2) partition, and (3) classification--- iteratively.

Velocity-dependent KAM islands

We revisit the non-dissipative time-dependent annular billiard and we consider the chaotic dynamics in two planes of conjugate variables in order to describe the behavior of the growth, or saturation, of the mean velocity of an ensemble of particles. We observed that the changes in the 4-d phase space occur without changing any parameter. They occur depending on where the initial conditions start. The emerging KAM islands interfere in the behavior of the particle dynamics especially in the Fermi acceleration mechanism. We show that Fermi acceleration can be suppressed, without dissipation, even considering the non-dissipative energy context. (C) 2011 Published by Elsevier Ltd.

Velocity-dependent KAM islands

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)