1. Order and chaos in some Hamiltonian systems of interest in plasma physics
Boris Weyssow
Universite Libre de Bruxelles
Dana Constantinescu
University of Craiova, Romania
Emilia Petrisor, University of Timisoara, Romania
Jacques Misguich, CEA Cadarache, France

2. A class of Hamiltonian systems is studied in order to describe, from a mathematical point of view, the structure of the magnetic field in tokamaks with reversed shear configuration.
The magnetic transport barriers are analytically located and described for various safety factors and perturbations.
General explanations for some experimental observations concerning the transport
barriers are issued from the analytical properties of the models:
-   the transport barriers are obtained in the presence of a reversed magnetic shear
in the negative or low shear region
(Litaudon X (1998), Maget P (2003), Neudatchin S. V. (2004))
-     zones with reduced transport appear when the minimum value of the safety factor closed to a low rational number
(Lopez Cardozo N.J. (1997), Gormezano C. (1999),
Garbet X. (2001), Jofrin E. (2002), Neudatchin S. V.(2004)

3. Tokamaks are toroidal devices used in thermocontrolled nuclear fusion
JET (EUR)
Magnetic field
Toroidal Poloidal
component component
helical magnetic field lines on nested tori sorrunding the magnetic axis (the ideal case)

4. The magnetic field equations Hamiltonian system
Toroidal coordinates
 = toroidal angle
= polar coordinates in a poloidal cross-section
The magnetic field equations Hamiltonian system
Clebsch representation
is the magnetic field
is the poloidal flux
is the toroidal flux
Unperturbed case:
regular (helical) magnetic lines
chaotic+regular magnetic lines
Perturbed case

5. is an area-preserving map compatible with the toroidal geometry
The discrete system
( Poincare map associated with the poloidal cross-section )
is an area-preserving map compatible with the toroidal geometry
(the magnetic axis is invariant)
(because )
is the winding function
is the safety factor (the q-profile)
is the magnetic shear
is the perturbation
K is the stochasticity parameter

6. the tokamap, R. Balescu (1998)
Chirikov-Taylor (1979)
Wobig (1987)
D. Del Castillo Negrete (1996)
the tokamap, R. Balescu (1998)
the rev-tokamap, R. Balescu (1998)

7. D. del Castillo Negrete, Greene J.M.,Morrison P.J. (1996,1997)
Area-preserving maps
Non-twist:
Twist :
(Monotonous winding function)
(positive or negative shear)
(Non monotonous winding function)
(reversed shear)
Poincare H. (1893) foundation of dynamical systems theory
Birkhoff G. D, ( ) fundamental theorems
KAM (Moser 1962) (persistence of invariant circles)
Greene J. M,
Aubry M & Mather J. N.
MacKay R. S. Percival I. C.( )
(break-up of invariant circles,converse KAM theory etc)
D. del Castillo Negrete, Greene J.M.,Morrison P.J. (1996,1997)
(routes to chaos in standard map systems)
Delshams A., R. de la Llave (2000)
(KAM theory for non-twist maps)
Simo C. (1998) (invariant curves in perturbed n-t maps)
Petrisor E. (2001, 2002) (n-t maps with symmetry group, reconnection)

8. Robust invariant circles (ITB) separating two invariant chaotic zones
The rev-tokamap
K=6.21
K=3.5
K=4.5
K=5.5
Robust invariant circles (ITB) separating two invariant chaotic zones

9. Rev-tokamap is a non-twist map
(the critical twist circle)
The nontwist annulus (NTA) is the closure of all orbits starting from the critical twist circle.

10. The revtokamap is closed to an almost integrable map in an annulus surrounding the curve
NTA contains the most robust invariant circles
The magnetic transport barrier surrounds the shearless curve
A magnetic transport barrier appears near 0 shear curve,
even in systems involving monotonous q-profile

11. ITB (the physical transport barrier)
For K< twist invariant circles
exist in the upper part of ITB
For K> all invariant circles in the upper part of ITB are nontwist
K=1.6
K=1.7

12. The destruction of invariant circles
Unbounded component in the negative twist region
Bounded component in the positive twist region
Theorem A A
No invariant circle pass through the points of
No invariant circle pass through the points of
K=0.5
K=4.1375 A A
as long as A belongs to the negative twist region.
K=5
K=6
is the intersection of
with the line

13. Reconnection phenomena
(global bifurcation of the invariant manifolds of regular hyperbolic points
of two Poincare-Birkhoff chains with the same rotation number)
Before reconnection: heteroclinic connections
between the hyperbolic points in each chain
Reconnection:
-connections between the hyperbolic points of distinct chains
-heteroclinic connections in the same chain
After reconnection: homoclinic+heteroclinic
connections in each chain
The chains are separated by meanders
Theorem
The reconnection of twin Poincare-Birkhoff chains
occurs in the NTA

14. Scenario for reconnection
(the same perturbation, modified W)
for w>n/m the two P-B chains of type (n,m)
are outside NTA
w decreases
the P-B chains enter NTA but they are
still separated by rotational circles
w decreases
the hyperbolic points reconnect
w decreases
the meanders separate the P-B chains
having homoclinic connection
w decreases
the first collision-annihilation occurs
w=0.53
w=0.51
w decreases
the second collision-annihilation occurs
w decreases
there are no more periodic orbits
of type (n,m)
w=0.50
w=0.49

15. Conclusions
The rev-tokamap model was used for the theoretical study of magnetic transport barriers
observed in reversed shear tokamaks.
Analytical explanations were proposed for
-the existence of transport barriers in the low shear regions
-the enlargement of the transport barriers when the minimum value of the q-profile
is closed to a low order rational.
Results:
A magnetic transport barrier appears near 0 shear curve, even in systems involving
monotonous q-profile.
In the rev-tokamap model the shape of the winding function has only quantitative
importance in the size of NTA.
The enlargement of NTA is directly related to the maximum value of the winding
function (corresponding to the minimum value of the safety factor).