Self-consistent chaotic transport in a high-dimensional mean-field Hamiltonian map model

TL;DR

This paper investigates self-consistent chaotic transport in a high-dimensional Hamiltonian mean-field model, revealing the formation of coherent structures and the role of periodic orbits in transport mechanisms.

Contribution

It introduces a novel high-dimensional mean-field map model with dynamical perturbation variables and analyzes the impact of periodic orbits on chaotic transport.

Findings

Self-consistency induces a macro-particle trapped around an elliptic fixed point.

A non-autonomous map models the asymptotic state and onset of global transport.

Sequential periodic orbits can be generated and continued from low-dimensional orbits.

Abstract

Self-consistent chaotic transport is studied in a Hamiltonian mean-field model. The model provides a simplified description of transport in marginally stable systems including vorticity mixing in strong shear flows and electron dynamics in plasmas. Self-consistency is incorporated through a mean-field that couples all the degrees-of-freedom. The model is formulated as a large set of $N$ coupled standard-like area-preserving twist maps in which the amplitude and phase of the perturbation, rather than being constant like in the standard map, are dynamical variables. Of particular interest is the study of the impact of periodic orbits on the chaotic transport and coherent structures. Numerical simulations show that self-consistency leads to the formation of a coherent macro-particle trapped around the elliptic fixed point of the system that appears together with an asymptotic periodic behavior of the mean field. To model this asymptotic state, we introduced a non-autonomous map that allows a detailed study of the onset of global transport. A turnstile-type transport mechanism that allows transport across instantaneous KAM invariant circles in non-autonomous systems is discussed. As a first step to understand transport, we study a special type of orbits referred to as sequential periodic orbits. Using symmetry properties we show that, through replication, high-dimensional sequential periodic orbits can be generated starting from low-dimensional periodic orbits. We show that sequential periodic orbits in the self-consistent map can be continued from trivial (uncoupled) periodic orbits of standard-like maps using numerical and asymptotic methods. Normal forms are used to describe these orbits and to find the values of the map parameters that guarantee their existence. Numerical simulations are used to verify the prediction from the asymptotic methods.