Nontwist maps

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Area-preserving nontwist maps are low-dimensional models of physical systems that locally violate the twist condition. An important problem for applications is the determination and understanding of the transition to global chaos (global transport) in these models. Nontwist maps exhibit several different phenomena: the break-up of invariant tori and separatrix reconnections. The latter may or may not lead to global transport, depending on the region of parameter space.

Apart from their physical importance, nontwist maps are of mathematical interest because they do not satisfy the usual assumptions of the celebrated Poincaré-Birkhoff theorem, the KAM theorem, and Aubry-Mather theory, and thus new results concerning typical behavior are required; however, a version of the KAM theorem has been extended to nontwist maps. (Delshams and de la Llave, 2000)

The composition of two twist maps does not necessarily result in another twist map, as noted by Moser (see e.g. 1986) when he introduced the twist condition. Generically, nontwist regions have been shown to appear in a neighborhood of an elliptic fixed point of an area-preserving map undergoing tripling bifurcation. (van der Wheele et al., 1988; Moeckel, 1990; Dullin et al., 2000)

The most studied example is the standard nontwist map (SNM), introduced in del-Castillo-Negrete and Morrison (1993). It is defined by \( \left(x,y\right)\in {\mathbb T}\times {\mathbb R}\rightarrow \left(\bar{x},\bar{y}\right)\in {\mathbb T}\times {\mathbb R}\) (with \({\mathbb T}\) the one-torus), where \[\tag{1} \begin{array}{lcl} \bar{x} & = & x + a \left( 1-\bar{y}^2\right),\quad \mbox{(modulo 1)}\\ \bar{y} & = & y -b \sin\left(2\pi x\right) \end{array} \]

and \(a,b\in{\mathbb R}\) are parameters. This map is area-preserving and violates the twist condition, \[\tag{2} \left|\frac{\partial \bar{x}\left(x,y\right)}{\partial y}\right| \geq c > 0\qquad \forall \ (x,y)\,, \]

along a curve in \((x,y)\)-space. In general, the twist condition Eq.(2) can be violated in various ways. A "nontwist" map violates the twist condition locally in a generic way with a simple traversal of zero in contrast to maps that are globally degenerate (sometimes referred to as "no-twist" maps) or maps that have traversals with vanishing derivatives, or tangencies of higher order. Although the SNM is not generic due to its symmetries, it captures the essential features of nontwist systems with a local, approximately quadratic extremum of the winding number profile.

One consequence of the violation of the twist condition is the existence of multiple periodic orbits with the same winding number which are already present in the integrable limit of nontwist maps. Changing the map parameters a and b causes bifurcations of periodic orbit chains with the same winding number: orbits can undergo chaotic layer reconnection ("separatrix" reconnection), or they can collide and annihilate.

Few results are known for symplectic nontwist maps in higher dimensions. A first step has been taken by Dullin and Meiss (2003) who have investigated twist singularities that occur in the neighborhood of an elliptic fixed point of four-dimensional symplectic maps using a normal form approach. Independently, Laskar has investigated higher dimensional maps using the method of frequency map analysis. (see e.g. Laskar, 2003)

Applications

There are many applications of nontwist systems. A nonexhaustive selection of references is given here to provide an entry-way into the literature. Historically two major fields of application of nontwist maps have been:

1. The modeling of chaos and reconnection in magnetic fields of toroidal plasma devices, such as tokamaks, that have reversed magnetic shear.(del-Castillo-Negrete and Morrison, 1992a; Oda and Caldas, 1995; Balescu, 1998)

In tokamaks, plasmas are confined by a magnetic field with helically shaped magnetic field lines that lie on tori, whose pitch or winding number is a function of position that is conventionally referred to as the safety factor or \(q\)-profile. As a low order approximation, the charged particles forming the plasma (or more precisely their guiding centers) stream along the field lines and hence the topology of the field lines is a dominant factor that determines plasma confinement. Reversed magnetic shear experiments have a \(q\)-profile that possesses a local minimum, \(q_{min}\ .\) Evidence in such experiments suggests the existence of a transport barrier around \(q_{min}\ ,\) resulting in enhanced confinement. The equations that describe magnetic field lines are a Hamiltonian system, and therefore the spatial behavior of magnetic field lines in tokamaks can be studied by means of a Poincaré map. The field line structure of reversed magnetic shear configurations can be described by an area-preserving nontwist map. The shearless curve (see below) of a nontwist map model provides the transport barrier. Of particular interest is the determination of the parameter ranges for which the invariant tori and, in particular, the robust shearless curve exist, and hence prevents global transport.

2. The modeling of transport by traveling waves in shear flows with zonal flows, i.e., with nonmonotonic velocity profiles. (del-Castillo-Negrete and Morrison, 1992b,1993)

This problem is of particular interest in the edge of tokamaks and in geophysical fluid dynamics, where wave propagation in strong global shear flows, which are known generically as jets or zonal flows is an ubiquitous phenomenon. Examples are the Gulf stream and polar night jet above Antarctica. Because of planetary rotation, these flows are predominately two-dimensional, with the altitude being the ignorable dimension. The motion of fluid particles in a two-dimensional incompressible flow is mathematically analogous to a one degree-of-freedom, possibly time-dependent, Hamiltonian dynamics problem, with the stream function playing the role of the Hamiltonian, and the spatial coordinates playing the role of canonically conjugate variables. To leading order the zonal flows are directed along the lines of latitude, either eastward or westward, with variation in the flow speed along lines of longitude. This variation possesses a maximum, giving rise to transport phenomena, including barriers, described by nontwist Hamiltonian systems. The maximum in the flow profile corresponds to the shearless curve.

Below is a list of other selected applications of nontwist Hamiltonian systems (maps and flows):

RF acceleration in particle accelerators (Symon and Sessler, 1956)
Corrections to Keplerian orbits due to oblateness of planets (Kyner, 1968)
Laser-plasma coupling (Langdon and Lasinsky, 1975)
Magnetic field line structure in the double tearing mode (Stix, 1976)
Wave-particle interactions (Karney, 1978; Howard et al., 1986)
Beam-beam interaction in a storage ring (Gerasimov et al., 1986)
Transport and mixing in traveling waves (Weiss, 1991)
Ray propagation in a waveguide with a circular cross section and with a periodic array of lenses along its axis (Abdullaev, 1994)
Superconducting quantum interference devices (SQUIDs) (Kaufman et al., 1996)
Relativistic dynamics of periodically driven oscillators (Kim and Lee, 1995; Luchinsky et al., 1996b)
Magnetic field lines in stellerators (Davidson et al., 1995; Hayashi et al., 1995)
\(E\times B\) transport in magnetized plasmas (Horton et al., 1998; del-Castillo-Negrete, 2000) ))
Circular billiards (Kamphorst and de Carvalho, 1999; Howard, 2009)
Coherent structures and self-consistent transport in a Hamiltonian mean-field model for point-vortex interactions (del-Castillo-Negrete and Firpo, 2002)
Atomic physics (Chandre et al., 2002)
Stellar pulsations (Munteanu et al., 2002)

Properties of nontwist maps

Nontwist maps exhibit several features not found in twist maps.

Shearless invariant tori
Meandering tori
Separatrix reconnection

Shearless invariant tori

One of the main characteristics of nontwist maps, in comparison to twist maps, is that more than one rotational invariant circle, and more than one rotational chain of periodic orbits of the same winding number (rotation number) can exist. In the integrable standard nontwist map (the map Eq.(1) with \(b=0\)), for example, the invariant circles \(y=y_0, y=-y_0, y_0\in{\mathbb R}\ ,\) have the same rotation number \(\omega\left(y_0\right)=a\left(1-y_0^2\right)\ .\)

Multiple rotational orbits of the same winding number, \(\omega\ ,\) can collide with and annihilate each other under suitable changes in parameter values. (In the integrable case, two rotational orbits of rotation number \(\omega<a\) approach each other as \(a\) decreases and collide at \(a=\omega\)). Computer experiments revealed that when two rotational invariant circles of the same rotation number collide, the winding number profile exhibits a local extremum point (Figure 2) and the resulting invariant set at collision is referred to as the shearless invariant torus (del-Castillo-Negrete et al., 1996). (In the context of integrable normal forms this torus has also been referred to as the twistless circle ( Dullin et al., 2000)).

Figure 1: The phase portrait of the standard nontwist map corresponding to a = 0.615,b = 0.4.

Figure 2: Winding number profile along y-axis at a = 0.615, b = 0.4. The bottom of the dip in the upper left corner of the winding number profile correspond to the winding number of the shearless curve (shown in red in Figure 1). The top plateau in the profile corresponds to the 3/5 periodic orbit chains after reconnection.

As the parameters of the map are changed (in particular when increasing b) more and more rotational invariant tori are destroyed, but numerical studies have discovered that shearless curves can be remarkably robust.

Rotational invariant tori at breakup exhibit scale invariance under specific (x,y)-space re-scalings, which are observed to be universal for certain classes of area-preserving maps. Detailed studies of the breakup of shearless tori in nontwist maps (del-Castillo-Negrete et al., 1996,1997; Fuchss et al.,2006, 2007) have shown significant differences compared to the breakup of invariant tori in twist maps. (MacKay, 1983)

The self-similar structure of the rotational invariant tori at breakup is reminiscent of a phase transition. To explore this analogy, a theory for renormalization in area-preserving maps has been developed, originally in the context of twist maps (Kadanoff, 1981; Shenker and Kadanoff, 1982, MacKay, 1983; Greene, 1990) and later extended to nontwist maps. (del-Castillo-Negrete et al., 1997, Apte et al., 2005) In this theory, there is a renormalization group operator defined on the space of area-preserving maps whose critical fixed points correspond to maps that have an invariant torus at breakup.

Separatrix reconnection and meandering tori

Under suitable changes of parameter values, rotational periodic orbits of the same winding number approach each other in phase space and interact through chaotic layer reconnection ("separatrix" reconnection). The most generic example of reconnection is shown in Figure 3.

At a threshold the invariant manifolds of distinct hyperbolic periodic orbits, of the same period, intersect each other, as shown in Figure 3b. Further increase of the perturbation leaves each hyperbolic orbit with a homoclinic and a heteroclinic manifold (Figure 3c). In the region between the two chains, new periodic orbits and invariant tori appear. These tori are not graphs over the x-axis and have been called meanders or meandering curves. (van der Weele et al., 1988; Simo, 1998) Such invariant tori can occur only in nontwist maps because any invariant torus in a twist map must be a graph over x (Birkhoff's theorem). As the perturbation is increased further, the periodic orbits collide (Figure 3d) and are survived by the rotational meandering tori (Figure 3e).

Depending on the region in parameter space, the formation of meanders can be suppressed and the reconnection itself can lead to global transport.

Figure 3: Generic reconnection scenario in (x,y)-space as explained in the text. For fixed a the parameter b increases from left to right causing the reconnection and collision of the rotational periodic orbit chains. After the reconnection new rotational invariant meandering tori appear (green: shearless curve).

In addition to the example in Figure 3, the standard nontwist map can exhibit several other types of reconnection scenarios. (Wurm et al., 2005)

One example of reconnection which occurs in nontwist maps with symmetries is the formation of vortex pairs, as shown in Figure 4.

Figure 4: Nongeneric reconnection scenario in (x,y)-space. For fixed a the parameter b increases from left to right causing the reconnection and collision of the rotational periodic orbit chains. During the reconnection process vortex pairs are formed. (green: shearless curve).

Historical overview

Before a systematic study of nontwist Hamiltonian systems was undertaken, the phenomenon of separatrix reconnection was observed in a numerical study of rf acceleration in particle accelerators by Symon and Sessler (1956). Reconnection scenarios had also been conjectured early on by Stix (1976) in the context of the evolution of magnetic surfaces in the nonlinear double-tearing instability, and were found by Gerasimov et al. (1986) in a two-dimensional model of the beam-beam interaction in a storage ring.

Systematic studies of nontwist systems were undertaken by several groups in different contexts, often independent of each other and under different headlines:

Howard and Hohs (1984) presented a map equivalent to (1) (referred to as a radial twist map) and studied numerically the reconnection of low-order resonances exhibiting homoclinic and vortex-like structures. Defining an averaged Hamiltonian they predicted the reconnection threshold for period-one and period-two fixed points. Howard and Humpherys (1995) extended the study to cubic and quartic nontwist maps.
Degenerate resonances were studied by Morozov and Shilnikov (1983) in non-conservative systems close to two-dimensional Hamiltonian ones.
Reconnection in area-preserving nontwist maps was studied by van der Weele et al.(1988, 1990) in the context of area-preserving maps with a quadratic extremum. The terminology ``nontwist" and ``meanders" seems to have originated here. In addition these authors studied the effect of dissipation on the reconnection process in (van der Weele et al., 1990).
Generic bifurcation of the twist coefficient in area-preserving maps was studied by Moeckel (1990).
Integrable overlap of resonances were studied in the work of Carvalho and Almeida (1992) in nontwist Hamiltonian systems under integrable perturbations.
The importance of the shearless curve as a transport barrier in physical models described by nontwist Hamiltonians was realized by del-Castillo-Negrete and Morrison in (del-Castillo-Negrete and Morrison, 1992a) in the context of non-monotonic \(q\)-profiles in magnetic fusion devices and in (del-Castillo-Negrete and Morrison, 1992b, 1993) in the context of transport by Rossby waves in shear flows. These references mark the first appearance of the standard nontwist map in the form of Eq. (1). These authors studied the detailed breakup of a particular shearless torus with golden mean winding number using Greene's residue criterion in (del-Castillo-Negrete et al. (1996,1997), and the renormalization framework was extended to include nontwist systems. (del-Castillo-Negrete et al., 1997)
Nonlinear resonances of oscillators whose frequency of oscillation possess an extremum as a function of energy have been studied by Soskin (Soskin, 1994; Soskin and Luchinsky, 1995) under the name of zero-dispersion nonlinear resonances. Luchinsky et al. (1996a) extended this work to weakly dissipative systems.
Dullin et al. (2000) showed that nontwist regions appear generically in area-preserving maps that have a tripling bifurcation of an elliptic fixed point.
Breakup of shearless invariant tori and renormalization in nontwist Hamiltonian flows was studied by Gaidashev and Koch (2004).

References

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Nontwist maps

Contents

Applications

Properties of nontwist maps

Shearless invariant tori

Separatrix reconnection and meandering tori

Historical overview

Recommended reading

References

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