scholarly journals Hamiltonian Maps for Heliac Magnetic Islands

1995 ◽  
Vol 48 (5) ◽  
pp. 871 ◽  
Author(s):  
MG Davidson ◽  
RL Dewar ◽  
HJ Gardner ◽  
J Howard
Keyword(s):  
Magnetic Field ◽  
Field Line ◽  
Chaos Theory ◽  
Magnetic Islands ◽  
Twist Map ◽  
Hamiltonian Chaos ◽  
Area Preserving Maps ◽  
Vacuum Fields ◽  
Rotational Transform ◽  
Area Preserving

Magnetic islands in toroidal heliac stellarator vacuum fields are explored with Hamiltonian chaos theory and the associated area-preserving maps. Magnetic field line island chains are examined first analytically, with perturbation theory, and then numerically to produce Poincaré sections, which are compared with H−1 Heliac stellarator puncture plot diagrams. Rotational transform profiles are chosen to permit the comparison of twist map and nontwist map predictions with field line behaviour computed by a field line tracing computer program and observed experimentally.

scholarly journals On reversible maps and symmetric periodic points

2016 ◽  
Vol 38 (4) ◽  
pp. 1479-1498
Author(s):  
JUNGSOO KANG
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Periodic Points ◽  
Analogous Statement ◽  
Twist Map ◽  
Planar Domains ◽  
Area Preserving Maps ◽  
Symmetric Periodic Orbits ◽  
Area Preserving ◽  
Double Symmetries

In reversible dynamical systems, it is of great importance to understand symmetric features. The aim of this paper is to explore symmetric periodic points of reversible maps on planar domains invariant under a reflection. We extend Franks’ theorem on a dichotomy of the number of periodic points of area-preserving maps on the annulus to symmetric periodic points of area-preserving reversible maps. Interestingly, even a non-symmetric periodic point guarantees infinitely many symmetric periodic points. We prove an analogous statement for symmetric odd-periodic points of area-preserving reversible maps isotopic to the identity, which can be applied to dynamical systems with double symmetries. Our approach is simple, elementary, and far from Franks’ proof. We also show that a reversible map has a symmetric fixed point if and only if it is a twist map which generalizes a boundary twist condition on the closed annulus in the sense of Poincaré–Birkhoff. Applications to symmetric periodic orbits in reversible dynamical systems with two degrees of freedom are briefly discussed.

Almost-invariant surfaces for magnetic field-line flows

1996 ◽  
Vol 56 (2) ◽  
pp. 361-382 ◽  
Author(s):  
S. R. Hudson ◽  
R. L. Dewar
Keyword(s):  
Magnetic Field ◽  
Field Line ◽  
Magnetic Field Line ◽  
Gradient Flow ◽  
Invariant Surface ◽  
Infinite Dimensional ◽  
Invariant Surfaces ◽  
Magnetic Surfaces ◽  
Rotational Transform ◽  
The Magnetic Field

Two approaches to defining almost-invariant surfaces for magnetic fields with imperfect magnetic surfaces are compared. Both methods are based on treating magnetic field-line flow as a 1½-dimensional Hamiltonian (or Lagrangian) dynamical system. In thequadratic-flux minimizing surfaceapproach, the integral of the square of the action gradient over the toroidal and poloidal angles is minimized, while in theghost surfaceapproach a gradient flow between a minimax and an action-minimizing orbit is used. In both cases the almost-invariant surface is constructed as a family of periodic pseudo-orbits, and consequently it has a rational rotational transform. The construction of quadratic-flux minimizing surfaces is simple, and easily implemented using a new magnetic field-line tracing method. The construction of ghost surfaces requires the representation of a pseudo field line as an (in principle) infinite-dimensional vector and also is inherently slow for systems near integrability. As a test problem the magnetic field-line Hamiltonian is constructed analytically for a topologically toroidal, non-integrable ABC-flow model, and both types of almost-invariant surface are constructed numerically.

Hamiltonian Chaos

2019 ◽  
pp. 154-176
Author(s):  
David D. Nolte
Keyword(s):  
Quantum Chaos ◽  
Kam Theory ◽  
Rigid Rotator ◽  
Standard Map ◽  
Web Map ◽  
Classical Chaos ◽  
Hamiltonian Chaos ◽  
Area Preserving Maps ◽  
Area Preserving ◽  
The Web

Nondissipative or Hamiltonian systems are also capable of chaos as phase space volume is twisted and folded in area-preserving maps like the Standard Map. When nonintegrable terms are added to a potential function, Hamiltonian chaos emerges. The Standard Map (also known as the Chirikov map) for a periodically kicked rigid rotator provides a simple model with which to explore the emergence of Hamiltonian chaos as well as the KAM theory of islands of stability. A periodically kicked harmonic oscillator displays extended chaos in the web map. Hamiltonian classical chaos makes a direct connection to quantum chaos, which is illustrated using the chaotic stadium, for which quantum scars are associated with periodic classical orbits in the stadium.

scholarly journals Three-dimensional magnetohydrodynamic equilibria with continuous magnetic fields

2017 ◽  
Vol 83 (4) ◽  
Author(s):  
S. R. Hudson ◽  
B. F. Kraus
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Three Dimensional ◽  
Energy Functional ◽  
Equilibrium States ◽  
Equilibrium Solutions ◽  
Magnetic Islands ◽  
Continuity Properties ◽  
Rotational Transform ◽  
The Magnetic Field

A brief critique is presented of some different classes of magnetohydrodynamic equilibrium solutions based on their continuity properties and whether the magnetic field is integrable or not. A generalized energy functional is introduced that is comprised of alternating ideal regions, with nested flux surfaces with an irrational rotational transform, and Taylor-relaxed regions, possibly with magnetic islands and chaos. The equilibrium states have globally continuous magnetic fields, and may be constructed for arbitrary three-dimensional plasma boundaries and appropriately prescribed pressure and rotational-transform profiles.

scholarly journals A Topological Approach to Bend-Twist Maps with Applications

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
Anna Pascoletti ◽  
Fabio Zanolin
Keyword(s):  
Fixed Point Theorems ◽  
Small Perturbations ◽  
Topological Approach ◽  
Twist Map ◽  
Existence And Multiplicity ◽  
Twist Maps ◽  
Area Preserving Maps ◽  
Planar Systems ◽  
Twist Theorem ◽  
Area Preserving

In this paper we reconsider, in a purely topological framework, the concept of bend-twist map previously studied in the analytic setting by Tongren Ding in (2007). We obtain some results about the existence and multiplicity of fixed points which are related to the classical Poincaré-Birkhoff twist theorem for area-preserving maps of the annulus; however, in our approach, like in Ding (2007), we do not require measure-preserving conditions. This makes our theorems in principle applicable to nonconservative planar systems. Some of our results are also stable for small perturbations. Possible applications of the fixed point theorems for topological bend-twist maps are outlined in the last section.

Evolution of Large-Scale Coronal Structures

1994 ◽  
Vol 144 ◽  
pp. 29-33
Author(s):  
P. Ambrož
Keyword(s):  
Magnetic Field ◽  
Field Line ◽  
Large Scale ◽  
Coronal Magnetic Field ◽  
Source Surface ◽  
Solar Cycles ◽  
Characteristic Relationship ◽  
The Magnetic Field ◽  
Coronal Structures

AbstractThe large-scale coronal structures observed during the sporadically visible solar eclipses were compared with the numerically extrapolated field-line structures of coronal magnetic field. A characteristic relationship between the observed structures of coronal plasma and the magnetic field line configurations was determined. The long-term evolution of large scale coronal structures inferred from photospheric magnetic observations in the course of 11- and 22-year solar cycles is described.Some known parameters, such as the source surface radius, or coronal rotation rate are discussed and actually interpreted. A relation between the large-scale photospheric magnetic field evolution and the coronal structure rearrangement is demonstrated.

scholarly journals Effects of plasma turbulence on the nonlinear evolution of magnetic island in tokamak

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Minjun J. Choi ◽  
Lāszlo Bardōczi ◽  
Jae-Min Kwon ◽  
T. S. Hahm ◽  
Hyeon K. Park ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Nonlinear Evolution ◽  
Plasma Turbulence ◽  
Magnetized Plasmas ◽  
Tokamak Plasmas ◽  
Magnetic Islands ◽  
Magnetic Island ◽  
Reconnection Site ◽  
Ambient Turbulence

AbstractMagnetic islands (MIs), resulting from a magnetic field reconnection, are ubiquitous structures in magnetized plasmas. In tokamak plasmas, recent researches suggested that the interaction between an MI and ambient turbulence can be important for the nonlinear MI evolution, but a lack of detailed experimental observations and analyses has prevented further understanding. Here, we provide comprehensive observations such as turbulence spreading into an MI and turbulence enhancement at the reconnection site, elucidating intricate effects of plasma turbulence on the nonlinear MI evolution.

scholarly journals Hamiltonian Approach to Magnetic Fields with Toroidal Surfaces

1985 ◽  
Vol 40 (10) ◽  
pp. 959-967
Author(s):  
A. Salat
Keyword(s):  
Magnetic Field ◽  
Hamiltonian System ◽  
Magnetic Fields ◽  
Constant Pressure ◽  
Time Dependent ◽  
Hamiltonian Approach ◽  
One Dimensional ◽  
Mhd Equilibrium ◽  
Magnetic Surfaces ◽  
Rotational Transform

The equivalence of magnetic field line equations to a one-dimensional time-dependent Hamiltonian system is used to construct magnetic fields with arbitrary toroidal magnetic surfaces I = const. For this purpose Hamiltonians H which together with their invariants satisfy periodicity constraints have to be known. The choice of H fixes the rotational transform η(I). Arbitrary axisymmetric fields, and nonaxisymmetric fields with constant η(I) are considered in detail.Configurations with coinciding magnetic and current density surfaces are obtained. The approach used is not well suited, however, to satisfying the additional MHD equilibrium condition of constant pressure on magnetic surfaces.

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