# Self-Consistent Chaotic Transport in a High-Dimensional Mean-Field Hamiltonian Map Model

## Abstract

We studied the self-consistent chaotic transport in a Hamiltonian mean-field model. This 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. Furthermore, 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,more »

- Authors:

- Applied Mathematics and Systems Research Inst., Mexico D.F. (Mexico)
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Fusion Energy Sciences (FES) (SC-24)

- OSTI Identifier:
- 1341549

- Grant/Contract Number:
- AC05-00OR22725

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Qualitative Theory of Dynamical Systems

- Additional Journal Information:
- Journal Volume: 14; Journal Issue: 2; Journal ID: ISSN 1575-5460

- Publisher:
- Springer

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; Self-consistent transport; Normal forms; Sequential periodic orbits; Single-wave model; PLASMAS

### Citation Formats

```
Martínez-del-Río, D., del-Castillo-Negrete, D., Olvera, A., and Calleja, R. Self-Consistent Chaotic Transport in a High-Dimensional Mean-Field Hamiltonian Map Model. United States: N. p., 2015.
Web. doi:10.1007/s12346-015-0168-6.
```

```
Martínez-del-Río, D., del-Castillo-Negrete, D., Olvera, A., & Calleja, R. Self-Consistent Chaotic Transport in a High-Dimensional Mean-Field Hamiltonian Map Model. United States. doi:10.1007/s12346-015-0168-6.
```

```
Martínez-del-Río, D., del-Castillo-Negrete, D., Olvera, A., and Calleja, R. Fri .
"Self-Consistent Chaotic Transport in a High-Dimensional Mean-Field Hamiltonian Map Model". United States. doi:10.1007/s12346-015-0168-6. https://www.osti.gov/servlets/purl/1341549.
```

```
@article{osti_1341549,
```

title = {Self-Consistent Chaotic Transport in a High-Dimensional Mean-Field Hamiltonian Map Model},

author = {Martínez-del-Río, D. and del-Castillo-Negrete, D. and Olvera, A. and Calleja, R.},

abstractNote = {We studied the self-consistent chaotic transport in a Hamiltonian mean-field model. This 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. Furthermore, 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.},

doi = {10.1007/s12346-015-0168-6},

journal = {Qualitative Theory of Dynamical Systems},

number = 2,

volume = 14,

place = {United States},

year = {2015},

month = {10}

}

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Works referencing / citing this record:

##
Hyperchaos in constrained Hamiltonian system and its control

journal, July 2018

- Li, Junhong; Wu, Huibin; Mei, Fengxiang
- Nonlinear Dynamics, Vol. 94, Issue 3