Universality in Three-Frequency Resonances

Julyan H. E. Cartwright; Diego L. Gonzalez; Oreste Piro

doi:10.1103/physreve.59.2902

Home/Archive/arXiv

arXiv · chao-dyn/9905034

Universality in Three-Frequency Resonances

Julyan H. E. Cartwright·Diego L. Gonzalez·Oreste Piro

View original source ↗Read source document ↗Export metadata ↓

Abstract

We investigate the hierarchical structure of three-frequency resonances in nonlinear dynamical systems with three interacting frequencies. We hypothesize an ordering of these resonances based on a generalization of the Farey tree organization from two frequencies to three. In experiments and numerical simulations we demonstrate that our hypothesis describes the hierarchies of three-frequency resonances in representative dynamical systems. We conjecture that this organization may be universal across a large class of three-frequency systems.

Explore related subjects

chao-dyn math.DS nlin.CD

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Julyan H. E. Cartwright, Diego L. Gonzalez, Oreste Piro. 1999-05-20. Universality in Three-Frequency Resonances. https://doi.org/10.1103/physreve.59.2902

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

KEEP EXPLORING

Related papers

arXiv2024

Dynamical ensembles equivalence in fluid mechanics

Dissipative Euler and Navier Stokes equations are discussed with the aim of proposing several experiments apt to test the equivalence of dynamical ensembles and the chaotic hypothesis.

chao-dyn↗

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↗