Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · chao-dyn/9805019

An Approximate KAM-Renormalization-Group Scheme for Hamiltonian Systems

C. Chandre·H. R. Jauslin·G. Benfatto
View original source ↗Read source document ↗Export metadata ↓

Abstract

We construct an approximate renormalization scheme for Hamiltonian systems with two degrees of freedom. This scheme is a combination of Kolmogorov-Arnold-Moser (KAM) theory and renormalization-group techniques. It makes the connection between the approximate renormalization procedure derived by Escande and Doveil, and a systematic expansion of the transformation. In particular, we show that the two main approximations, consisting in keeping only the quadratic terms in the actions and the two main resonances, keep the essential information on the threshold of the breakup of invariant tori.

Explore related subjects

chao-dynnlin.CD

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

C. Chandre, H. R. Jauslin, G. Benfatto. 1998-05-22. An Approximate KAM-Renormalization-Group Scheme for Hamiltonian Systems. https://doi.org/10.1023/a%3A1004519514702

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2015

The Bogdanov Map: Bifurcations, Mode Locking, and Chaos in a Dissipative System

We investigate the bifurcations and basins of attraction in the Bogdanov map, a planar quadratic map which is conjugate to the Hénon area-preserving map in its conservative limit. It undergoes a Hopf bifurcation as dissipation is added, and exhibits the panoply of mode locking, Arnold tongues, and chaos as an invariant circle grows out, finally to be destroyed in the homoclinic tangency of the manifolds of a remote saddle point. The Bogdanov map is the Euler map of a two-dimensional system of ordinary differential equations first considered by Bogdanov and Arnold in their study of the versal unfolding of the double-zero-eigenvalue singularity, and equivalently of a vector field invariant under rotation of the plane by an angle $2π$. It is a useful system in which to observe the effect of dissipative perturbations on Hamiltonian structure. In addition, we argue that the Bogdanov map provides a good approximation to the dynamics of the Poincaré maps of periodically forced oscillators.

chao-dyn↗
arXiv2012

Passive Scalars and Three-Dimensional Liouvillian Maps

Global aspects of the motion of passive scalars in time-dependent incompressible fluid flows are well described by volume-preserving (Liouvillian) three-dimensional maps. In this paper the possible invariant structures in Liouvillian maps and the two most interesting nearly-integrable cases are investigated. In addition, the fundamental role of invariant lines in organizing the dynamics of this type of system is exposed. Bifurcations involving the destruction of some invariant lines and tubes and the creation of new ones are described in detail.

chao-dyn↗