# Shearless effective barriers to chaotic transport induced by even twin
islands in nontwist systems

M. Mugnaine<EMAIL_ADDRESS>Institute of Physics, University of São Paulo,
São Paulo, SP, 05508-090, Brazil  I. L. Caldas  Institute of Physics,
University of São Paulo, São Paulo, SP, 05508-090, Brazil  J. D. Szezech Jr
Graduate in Science Program—Physics, State University of Ponta Grossa, Ponta
Grossa, PR, 84033-240, Brazil  Department of Mathematics and Statistics, State
University of Ponta Grossa, Ponta Grossa, PR, 84033-240, Brazi  R. L. Viana
Department of Physics, Federal University of Paraná, Curitiba, PR, 81531-980,
Brazil  P. J. Morrison  Department of Physics and Institute for Fusion
Studies, The University of Texas at Austin, Austin, TX 78712, United States

###### Abstract

For several decades now it has been known that systems with shearless
invariant tori, nontwist Hamiltonian systems, possess barriers to chaotic
transport. These barriers are resilient to breakage under perturbation and
therefore regions where they occur are natural places to look for barriers to
transport. Here we describe a novel kind of effective barrier that persists
after the shearless torus is broken. Because phenomena are generic, for
convenience we study the Standard Nontwist Map (SNM), an area-preserving map
that violates the twist condition locally in the phase space. The novel
barrier occurs in nontwist systems when twin even period islands are present,
which happens for a broad range of parameter values in the SNM. With a phase
space composed of regular and irregular orbits, the movement of chaotic
trajectories is hampered by the existence of both shearless curves, total
barriers, and a network of partial barriers formed by the stable and unstable
manifolds of the hyperbolic points. Being a degenerate system, the SNM has
twin islands and, consequently, twin hyperbolic points. We show that the
structures formed by the manifolds intrinsically depend on period parity of
the twin islands. For this even scenario the novel structure, named a torus
free barrier (TFB), occurs because the manifolds of different hyperbolic
points form an intricate chain atop a dipole configuration and the transport
of chaotic trajectories through the chain becomes a rare event. This structure
impacts the emergence of transport, the escape basin for chaotic trajectories,
the transport mechanism and the chaotic saddle. The case of odd periodic
orbits is different: we find for this case the emergence of transport
immediately after the breakup of the last invariant curve, and this leads to a
scenario of higher transport, with intricate escape basin boundary and a
chaotic saddle with non-uniformly distributed points.

##  1 Introduction

The analysis of a Hamiltonian system can be reduced to area preserving maps,
symplectic maps that arise as canonical Poincaré maps for systems of
differential equations [1, 2]. Near-integrable Hamiltonian systems possess a
phase space with coexistence of regular and irregular (chaotic) orbits. The
questions of where chaotic trajectories go and how long they take to move from
one region of the phase space to another naturally emerge, and are the subject
of a large literature on transport theory (see e.g. [3, 4]). Transport theory
concerns the collective movement of chaotic trajectories and includes
techniques for estimating the transport rates, which can be related to
quantities in experiments [5]. Transport analysis has application in many
research fields, such as plasma confinement [6, 7], fluid dynamics [8, 9], and
celestial mechanics [10, 11].

Hamiltonian systems possess an intermixture of periodic, quasi-periodic, and
chaotic orbits and the movement of the irregular components can be affected by
partial barriers and remnants of invariant tori, the Cantori [3, 4]. A
Cantorus is an invariant set with irrational frequencies formed by an infinite
number of gaps [6], which can provide a robust partial barrier in the phase
space. The existence of Cantori in the phase space has only been proven for
nondegenerated systems, i.e., systems where the frequency of the motion on a
torus is related with the action of the torus by a monotonic function (see
e.g. [9, 12]). Maps with this monotonic behavior are called twist maps, and
the KAM theorem and Aubry-Mather theory can be applied since the twist
condition, defined for maps as ${\partial x_{n+1}}/{\partial y_{n}}\neq 0$, is
satisfied at every point in the phase space [13, 14]. A map that violates such
condition is called nontwist and its universal behavior can be described by
the standard nontwist map (SNM) given in [9], the simplest mathematical model
that violates the twist condition.

Even though the existence of Cantori has not been proven for nontwist maps and
conventional KAM theorem cannot be applied at every point of the phase space,
there are rigorous KAM results for nontwist maps [15]. Moreover, transport has
been widely studied in nontwist systems (e.g. [16, 17, 18, 19, 20, 21, 22, 23,
24]). In the SNM, depending on parameter values, the phase space can be
composed of a chaotic sea bounded by invariant curves with two chains of
islands immersed in the chaotic regions. Due to the violation of the twist
condition, the two chains of islands are twins, i.e., have the same frequency
but are not connected [25, 26]. Among the invariant curves, there is the
shearless curve formed by the points where the twist condition is violated.
The shearless curve is also the extremum point of the nonmonotonic frequency
function. In a transport portrayal, the shearless curve acts as a barrier,
i.e., divides the chaotic sea into two regions and thereby preventing global
transport. Thus, the transition to global chaos is related to the destruction
of this barrier as was first investigated by Greene’s method in [13].

After the shearless curve breaks, chaotic orbits can be limited by the
existence of partial barriers. As noted in [19], there can be an effective
barrier formed by the remnants of the shearless curve, which trap the chaotic
orbits. There is also the phenomenon of stickiness around the twin islands.
The manifolds also have an important role in creating the partial barriers
[27, 16], since the intra-intercrossing scenario of the manifolds has a effect
on the transport intensity. Interestingly, the transport studies of these
earlier works were performed for the standard nontwist map with island chains
of odd period. Due to the map symmetries, the relative position of the islands
change for different parity. For the odd case, a hyperbolic and an elliptic
orbit are aligned, while for the even case two elliptic points are aligned.
Because there is no dynamical manifold based description for the scenario of
partial barriers with islands of even period, we provide numerical evidence
for characterizing the behavior. Our characterization of these partial
barriers yields an explanation for zero and low transport for certain
parameters of the SNM.

In our numerical survey of the SNM we analyze even and odd scenarios and
compare the transport mechanisms acting in the two cases. We show that for the
even scenario, the transport of chaotic trajectories is obstructed by a
network formed by the stable and unstable manifolds related to the chains of
hyperbolic points. We called this network structure a torus free barrier
(TFB), since there is no invariant curve, but the transport rarely happens due
to the manifold configuration. These results can be generalized for more
complex degenerate Hamiltonian systems with the same structures of twin island
chains.

Our paper is organized as follows: in Sec. 2 we present the standard nontwist
map and discuss features of the phase space related to the parity of the
islands. We also discuss the nonmonotonic behavior of the frequency function.
Transmissivity through the phase space is discussed in Sec. 3. Here we
investigate the escape of chaotic trajectories from the escape basin and
measure their escape times. In Sec. 4, we describe the TFB structure that is
responsible for the null/low transport of the system. Then, in Sec. 5 we
further analyze the transport through the TFB, explaining the rarity of such
transport of a chaotic trajectory. Finally, in Sec. 6 we conclude.

##  2 Parity scenarios for the standard nontwist map

The standard nontwist map is an area preserving map that was obtained from a
Hamiltonian model that describes transport and mixing properties of Rossby
waves in zonal shear flows [9]. It also arises from a model for the motion of
charged particles in drift waves in confined plasmas [7]. The map is given by

$\displaystyle\begin{aligned} y_{n+1}=&y_{n}-b\sin(2\pi x_{n}),\\\
x_{n+1}=&x_{n}+a(1-y_{n+1}^{2}),\hskip
20.00003pt\text{mod}~{}~{}1,\end{aligned}$ (1)

where the canonically conjugate variables are $y\in\mathbb{R}$ and
$x\in[0,1]$. The parameters $a$ and $b$ are real and independent of each
other, and the domain of interest is $a\in[0,1]$ and $b\in\mathbb{R}$. The map
violates the twist condition, $\partial x_{n+1}/\partial y_{n}\neq 0$, at
points in the phase space which belong to the so-called shearless curve. This
map was first thoroughly studied in [13, 28], where the breakup and scaling of
the shearless curve were analyzed.

For $b\neq 0$ the SNM represents a nonintegrable Hamiltonian system with a
phase space with coexistence of chaotic and regular solutions. Quasiperiodic
solutions, such as the shearless curve, can be represented by a curve that
spreads across the whole domain of $x$. There are also quasiperiodic orbits
that wind around any stable periodic solution. These orbits constitute the
islands and, due to the violation of the twist condition and the symmetry of
the map, they come in pairs adjacent to the shearless curve [28, 29].

The search for periodic solutions can be reduced to a one-dimensional problem
by the use of symmetries of the system, as first explained and used in [13].
In the SNM, the primary periodic stable orbits (elliptic points) are always on
one of the symmetry lines, defined by

$\displaystyle\begin{aligned} s_{0}=&\\{(x,y)|x=0\\},\\\
s_{1}=&\left\\{(x,y)|x={a(1-y^{2})}/{2}\right\\}.\end{aligned}$ (2)

Thus, the search for stable periodic orbits is a root finding problem along
these lines. The symmetry properties of the SNM are due to the symmetry
transformation $T_{S}(x,y)=\left(x+\frac{1}{2},-y\right)$, with which the
relation $M_{SNM}T_{S}=T_{S}M_{SNM}$, with $M_{SNM}$ defined by (1), is valid.
The symmetry transformation can be applied to the symmetry lines in (2), and
two other lines can be found:

$\displaystyle\begin{aligned} s_{2}=&\left\\{(x,y)|x={1}/{2}\right\\},\\\
s_{3}=&\left\\{(x,y)|x={a(1-y^{2})}/{2}+{1}/{2}\right\\}.\end{aligned}$ (3)

As mentioned, the periodic orbits in the map come in pairs, i.e., there are
two periodic solutions with the same frequency, one on each side of the
shearless curve [28]. According to the period parity of the periodic orbits,
their relative positions are different if the period is even or odd. For a
scenario where the periodic orbits exhibit even period, the fixed points of
the same stability, of each chain of islands, belongs to the same symmetry
line, while for odd period, the points with opposite stability are aligned
along the symmetry line. In order to illustrate this, we compute phase space
plots for the SNM with islands of period 2 and 3, along with the four symmetry
lines defined by (2) and (3). The plots are shown in Figure 1.

Figure 1: Phase space plots for the even and odd scenarios. In (a) and (c) we
choose $a=0.53$ and for (b) and (d), $a=0.71$. The islands are represented by
the colored closed curves, the hyperbolic points by the triangle and diamond
symbols, and the shearless curve, when it exists, by the thick curve. The even
scenario before the breakup of the shearless curve is shown in (a) for
$b=0.45$, while the odd scenario with the curve is represented in (b), for
$b=0.39$. In (c) and (d), we choose $b=0.53$ and $b=0.48$ in order to
represent the even and odd scenario after the breakup of the shearless curve,
respectively.

In Figure 1 we have phase spaces plots for the even and odd scenarios along
with the symmetry lines $s_{i}$, for $i=0,1,2$ and 3 from equations (2) and
(3). The symmetry lines $s_{0}$ and $s_{2}$ are the vertical-dashed lines at
$x=0$ and $x=0.5$ respectively, while the lines $s_{1}$ and $s_{3}$ correspond
to the left and right dashed parabolas. For the even scenario, we chose
$a=0.53$ and $b=0.45$ for panel (a) and $b=0.53$ for panel (c). In the phase
space plots for the odd scenario, we have $a=0.71$ and $b=0.39$ for panel (b)
and $b=0.48$ for panel (d).

In Figure 1 (a), the islands have period two, and they are indicated by the
colored closed curves. We indicate the islands as upper and lower island
chains with different colors: the upper (lower) chain in green (magenta) is in
the upper (lower) side of the shearless curve, represented by the thick black
curve in the center. In panel (a), we observe elliptic points on the same
symmetry lines $s_{0}$ and $s_{2}$ while the hyperbolic points, represented by
the triangle and diamond symbols, are also in the same line, but in this case
$s_{1}$ and $s_{3}$. The scenario for odd period is quite different. As seen
in Figure 1 (b), in the same symmetry line we have an elliptic and a
hyperbolic point.

When the shearless curve no longer exists, we have the phase space plots of
Figure 1 (c) and (d), for the even and odd scenarios, respectively. In Figure
1 (d), we observe the same behavior seen in Figure 1 (b) when the curve was
present: the elliptic and hyperbolic points are aligned along the symmetry
line. This is not observed for the even scenario. As long as the elliptic
points are still aligned in Figure 1 (c), the hyperbolic points are out of the
symmetry lines and, consequently, the two chains of hyperbolic points are no
longer related by the symmetry transformation $T_{S}$. The hyperbolic points
are out of the symmetry line due to the scattering that happens after the
collision of the points at the shearless breakup. For a detailed discussion
about the collision, see [13].

The thick curves in Figure 1 (a) and (b) are the shearless curve, the set of
points where the twist condition is violated and the winding number assumes an
extreme value. The winding number of an orbit, also called rotation number, is
defined by the limit,

$\displaystyle\omega_{n}=\lim_{n\to\infty}\dfrac{x_{n}}{n},$ (4)

which converges to a rational (irrational) value for a periodic
(quasiperiodic) orbit. The limit (4) does not converge if the orbit is
chaotic. Computing the winding number profiles along the symmetry line
$s_{2}:x=1/2$ for the phase space plots of Figure 1, we have the profiles
shown in Figure 2.

Figure 2: Winding number profiles computed along the symmetry line $x=0.5$ for
the phase space plots of Figure 1. The parameters $a$ and $b$ for (a)-(d) are
the same as those for Figure 1 (a)-(d), respectively. The red stars indicate
the maximum value of $\omega_{n}$ in (a) and (b). The magnification in (a) is
around the region framed by the dashed box.

For the winding number profiles of Figure 2, we compute the limit (4) for
$10^{5}$ initial conditions on the symmetry line $x=1/2$ with a final
iteration time $n=10^{6}$. Observing the phase space plots of Figure 1, the
majority of the solutions at the symmetry line are chaotic and they do not
have a defined $\omega_{n}$. For this reason, the domain of $y$ used for the
winding number profiles are reduced to $y\in[-0.4,0.4]$ for Figure 2 (a) and
(c), the even scenario, and $y\in[-0.1,0.4]$ for Figure 2 (b) and (d), the odd
scenario.

The even scenario with shearless curve is shown in Figure 2 (a), with $a=0.53$
and $b=0.38$. The profile $\omega_{n}$ is composed of two plateaus at
$\omega_{n}=0.5$ indicating the two islands of period 2. Between these two
plateaus, we observe nonmonotonic behavior of the winding number, highlighted
by the magnification: a parabola with the extreme point indicating the
position of the shearless curve. The points on the left and right of the
maximum point are due to other curves that occupy the whole domain of $x$ and
are not shown in Figure 1 (a). Similar behavior is observed in Figure 2 (b),
but in this case, we only observe one plateau at $\omega_{n}=1/3$,
representing the upper island of period 3. This difference is due to the
disposition of the islands in the symmetry lines as discussed before. Since
the elliptic points are aligned along the symmetry line for the even case, we
observe two plateaus, differently from the odd scenario where the hyperbolic
point is aligned with the elliptic point and only one plateau can be seen.

Figure 2 (c) and (d) represent the even and odd scenario for
$(a,b)=(0.53,0.53)$ and $(a,b)=(0.71,0.48)$, respectively. For these
parameters, the shearless curve no longer exists and in both profiles, only
the plateaus remain, indicating the persistence of the islands with the
increasing of the perturbation amplitude $b$. Once again, we can observe two
plateaus for the even case and just one for the odd scenario.

##  3 Transport across the phase space

As observed in Figure 1 (a) and (b), the shearless curve acts as a barrier to
transport, since it divides the chaotic sea into upper and lower regions.
Thus, the transition to global chaos and, consequently, global transport
requires the destruction of the shearless barrier [13]. For some set of
parameters $(a,b$) where the shearless curve no longer exists, chaotic
solutions can cross the phase space leading to high transport. In order to
quantify and describe the transport through the phase space, we use the
transmissivity quantifier and the concepts of escape time and escape basins.

###  3.1 Transmissivity

The transmissivity is a quantifier defined as the fraction of orbits that
cross a region of the phase space in a predefined time interval [19].
Numerically, we randomly choose $N_{T}=10^{6}$ initial conditions on the line
$y=-2.0$, iterate all of them for $100$ iterations and compute how many cross
the phase space and reach the line $y=2.0$. The fraction is the transmissivity
$T_{100}$. As performed before, we compute the quantifier for two scenarios:
even and odd. For the even scenario we again choose $a=0.53$ and for the odd,
$a=0.71$. The behavior of $T_{100}$ relative to the perturbation parameter $b$
is shown in Figure 3.

Figure 3: Transmissivity profile for the SNM with (a) $a=0.53$ representing
the even scenario and (b) $a=0.71$, representing the odd case. The red dashed
line indicates the $b$ value for the breakup of the shearless curve.

From the transmissivity profiles in Figure 3, we can affirm that transport
across phase space with islands of even period differs from that with islands
of odd period. The $b$ values for the breakup of the shearless curve are
indicated by the red dashed lines: $b\approx 0.4754$ and $b\approx 0.4635$ for
Figure 3 (a) and (b), respectively. In the even scenario, represented by the
profile in Figure 3 (a), we observe that even without a barrier, the transport
can be null for a large range of parameters for 100 iterations. The shearless
curve breaks at the point $b\approx 0.4754$ but a non-null transport is
noticed only for $b=0.70$. In contrast is the result observed in Figure 3 (b)
where the islands in the phase space have odd period. In this case, the
transport is non-null right after the shearless breakup. Therefore, for the
even scenario, we have a transport barrier without an actual barrier, a TFB.

In order to investigate the difference between no/low transport for the even
case and the higher transport for the odd case, we analyze the phase space and
the chaotic solutions for some values of $b$. We choose two values of $b$ for
the even case, one for the scenario with a TFB and another for the higher
transport observed in Figure 3 (a), therefore, $b=0.53$ and $b=0.80$,
respectively. For the odd case, we analyze the peak of transport for $b=0.494$
of the transmissivity profile of Figure 3 (b). We examine the three sets of
parameters using the concepts of escape basin and escape time, stated in the
next subsection.

###  3.2 Escape basins and escape time

Consider now how the chaotic solutions distributed over the phase space behave
during the time evolution of the map. For this, we unite the escape time and
the escape basin analysis, since we can set exits in the phase space of the
SNM. The escape basin associated with an exit is the set of points which
escape through this exit within a time interval, while the escape time is the
number of iterations a solution takes to go through the exit.

As in previous studies (see [16, 17, 18, 30]), we define two exits in the
phase space: the lines $y=\pm 1.0$. Then we uniformly distribute
$2.10^{3}\times 2.10^{3}$ initial conditions in the domain
$\mathcal{D}:\\{(x,y)|x\in[0,1]~{}\text{and}~{}y\in[-1,1]\\}$ and iterate all
of them for $10^{3}$ iterations. During the time evolution, if the solution
crosses one of the two exits $y=\pm 1$ we record which exit and also the
iteration number at which the crossing happens. We join these two pieces of
information in a color scale for the phase space. If the solution crosses the
exit $E^{+}:y=+1$ ($E^{-}:y=-1$), we set its initial condition as a green
(pink) point in the phase space. The shade of green and pink is related to the
escape time: the darker it is, the longer the escape time. The escape basins
and the escape times for $(a,b)=(0.53,0.53)$, $(a,b)=(0.53,0.80)$ and
$(a,b)=(0.71,0.494$) are show in Figure 4.

Figure 4: Escape time and escape basin for the SNM for the even scenario with
$a=0.53$, (a) $b=0.53$ and (b) $a=0.80$, and for the odd scenario, with (c)
$(a,b)=(0.71,0.494)$. The pink (green) points indicate the initial condition
of the solution which cross the exit $E^{-}:y=-1.0$ ($E^{+}:y=+1.0$) within
$10^{3}$ iterations. The color shade, as shown in the color bar, indicate the
escape time: light pink/green indicate a small escape time while the darker
points indicate longer escape times.

Figure 4 (a), with $(a,b)=(0.53,0.53)$ represents the even scenario with a
TFB, while Figure 4 (b) shows the peak of transport for the even scenario of
Figure 3 (a), with $(a,b)=(0.53,0.80)$. For both cases of Figure 4 (a) and
(b), we observe a smooth boundary between the green and the pink basins, i.e.,
there is no significant mixing between the two colored regions. Thus, the
trajectories which begin in the lower/upper region of the phase space tends to
escape through the lower/upper exit and, consequently, crossing the phase
space is a rare event. A different outcome is seen in the odd scenario of
Figure 4 (c), where the boundary between the basins is no longer smooth. In
fact, the boundary is intricate and there is a mixing region of the pink and
green points around the islands (the islands are indicated by the white
regions). This mixing leads to a greater uncertainty, since close initial
conditions can escape through different exits. From the result presented in
Figure 4 (c), we can also affirm that the crossing through the phase space is
more common, comparing to the case in Figure 4 (b), since there are green/pink
points in the lower/upper region.

From Figure 4, we observe that greater escape times happen for trajectories
with initial conditions near the islands. This is due to the stickiness
phenomenon, i.e, chaotic solutions get stuck around the quasi regular orbits
for a certain time interval. Figure 4 (a) presents the larger black region,
indicating a stronger stickiness in this case. For Figure 4 (b) and (c), we
observe a more restrained stickiness region around the islands. The majority
of the escape basins present light shades of pink and green, indicating a
short escape time. The darker shades are closer to the islands and,
apparently, distance themselves from the islands in patterns that resemble
manifolds. Therefore, it seems the manifolds reflect the behavior of the
chaotic solutions and their transport through the phase space.

##  4 Torus free barriers

Transport analyses aim to characterize the motion of chaotic trajectories
through phase space (e.g. [4]). Typically nonintegrable Hamiltonian systems
that are perturbations of integrable ones have a coexistence of chaos and
regularity, with islands and invariant curves that can affect, limit, and/or
preclude transport. The Poincaré-Birkhoff theorem states that for each
elliptic fixed point at the center of the islands there corresponds unstable
hyperbolic fixed points in the chaotic sea. As shown in [16, 19, 20, 27], the
stable and unstable manifolds related to the hyperbolic points impact the
transport of chaotic trajectories in the SNM. Following this idea in the
present context, we give here a manifold analysis for the phase space plots of
Figure 4, in order to explain the torus free barrier scenario and the peaks of
transmissivity through the phase space.

Following the notation of [16], let
$\mathbf{O^{U}}\left(\mathbf{O^{L}}\right)$ denote the hyperbolic points of
the upper (lower) island chain and denote its unstable and stable manifolds by
$W_{u}^{U}\left(W_{u}^{L}\right)$ and $W_{s}^{U}\left(W_{s}^{L}\right)$,
respectively. Because we are investigating a scenario with twin islands in
nontwist systems, the crossing between stable and unstable manifolds can form
an intracrossing scenario, where the crossings occur between manifolds of the
same chain of hyperbolic points, and the intercrossing scenario, where the
crossings are between the manifolds of different chains of hyperbolic points
[16, 19]. For the phase space plots of Figure 4, the stable and unstable
manifolds of each chain of hyperbolic points are shown in Figure 5.

Figure 5: Manifold structure for the phase space plots of Figure 4. The upper
(lower) hyperbolic points $\mathbf{O^{U}}$ ($\mathbf{O^{L}}$) are represented
by the triangle (diamond) symbol. The manifolds $W_{s}^{U}$, $W_{s}^{L}$,
$W_{u}^{U}$ and $W_{u}^{L}$ are indicated by the blue, green, orange and pink
curves. Close to the hyperbolic points, we draw arrows indicating the
stability direction: inwards and outwards, indicating the stable and unstable
directions of the hyperbolic point.

Figure 5 depicts the stable and unstable manifolds related to hyperbolic
points of the SNM for the same parameters as those of Figure 4. In Figure 5
(a) the manifolds are shown for the case of the TFB, where we have no
transport without a real barrier. The blue and pink curves, representing
$W_{s}^{U}$ and $W_{u}^{L}$, respectively, surround the two islands in a
dipole-like shape and the crossing structure between these two manifolds
resembles the usual turnstile/homoclinic structure. The intriguing and newly
observed structure is the one formed by the green and orange curves, the
stable and unstable manifolds of the lower and upper chain, $W_{u}^{U}$ and
$W_{s}^{L}$, respectively. These two manifolds are apparently superimposed and
form what appears to be a smooth curve across the interior of and between the
dipoles. This smooth curve correspond to the smooth boundary between the two
escape basins of Figure 4 (a). From our numerical simulations, we affirm that
there are no manifolds that connect different hyperbolic points in the region
inside and outside of the dipole. Even though it seems the manifolds occupy
the same region; in the region outside the dipole, the green curve oscillates
and moves away from the triangle hyperbolic point $\mathbf{O^{U}}$. The same
happens with the orange curve: it gets closer to the diamond point
$\mathbf{O^{L}}$, but then the manifold follows the pink manifolds and moves
away from the point. In the curve inside the dipole, there are small
oscillations and, consequently, crossings between the green and orange curves.
(The crossings are shown in the magnification given in Figure 6).

At the peak of transmissivity of Figure 3 (a) we have $(a,b)=(0.53,0.8)$; the
corresponding manifolds are presented in Figure 5 (b). We observe a structure
similar to the one presented in Figure 5 (a), but the crossing between the
orange and green curves are more evident. These crossings form lobes and
enable chaotic orbits to cross the phase space at these regions. Thus, the TFB
structure is modified and the transport through the phase space is higher.

For the odd scenario, presented in Figure 5 (c), we have the usual turnstile
structure. This is a well-known case already investigated and described in the
literature [4, 16]. The peak of transmissivity for this phase space structure
is due to the intercrossing between the manifolds of different chains of
hyperbolic points. For a more complete description of how the intercrossing is
responsible for an increase in the transport see [16].

##  5 Transport mechanism

As noted above, the manner in which trajectories traverse phase space for the
odd scenario case has been well-studied, including its understanding in terms
of conventional turnstiles. However, to our knowledge, the manner in which
trajectories cross the dipole-like configuration for the even scenario case
has not been previously investigated. For the even case when TFBs exist,
because they are not perfect barriers, transport can occur; however, our
numerical simulations show that this is rare and the number of iterations
needed is large. For example, for the phase space plot of Figure 5 (a), fewer
than $0.01\%$ of the trajectories cross the phase space in $10^{5}$
iterations. Thus, the TFB is a partial but quite strong transport barrier.

In order to understand the transport mechanism, we search for trajectories
that cross the phase space and analyze their behavior inside the dipole-like
structure. In particular, we analyze the transport through the structures
shown in Figure 5 (a) and (b). Our results are shown in Figure 6, where we
trace trajectory paths through the twin island chain region. We consider
representative initial conditions, highlighted by circles, and consider the
same parameters $a$ and $b$ of Figure 5 (a) and (b), respectively. In Figure 6
(a) an escape path is seen by following the arrows indicating the direction of
the trajectory. We observe that the trajectory crosses the TFB inside the
dipole-like configuration. In contrast, in Figure 6 (b) we see the crossing
happens in the region between the two dipoles. The magnifications of the
insets show how the crossing happens: the trajectory goes from the lower part
of the phase space to a region inside a lobe formed by the green and orange
manifold, and then stays inside the lobes upon each iteration. Then, due to
the crossings between these manifolds, the trajectory goes to the upper region
of the phase space. These scenaria occur for both phase space plots shown in
Figure 6.

Figure 6: Transport of a chaotic trajectory through the dipole-like
configuration. The initial condition of the orbit is highlighted with a
circumference and the direction of the orbit is indicated by arrows. In (a),
we have $b=0.53$ representing the scenario where the transport is a very rare
event while in (b), $b=0.8$, we have the peak of transmissivity of the profile
in Figure 3 (a). The manifold color scheme is the same as of Figure 5. The
insets in each panel are magnifications of the corresponding regions inside
the dashed squares.

As discussed in [3], there is a relationship between the area of the lobes
formed by the stable and unstable manifolds and the flux of trajectories
through phase space. Therefore, we can explain the low/high transport in the
phase spaces of Figure 6 (a) and (b), respectively, by the area of the lobes
formed by the manifolds. In Figure 6 (a), the lobes formed by the green and
orange manifolds are exceptionally small and can only be seen in the
magnifications. Furthermore, they are restricted to the region inside the
dipole region. The combination of small size with the restriction of being
only inside the dipoles is the reason the crossing of chaotic trajectories is
a rare event in this configuration. For the different escape path of Figure 6
(b), where the transport is higher, we can observe the lobes formed by the
crossings without any magnification, i.e., they are significantly bigger in
size. Moreover, there are significantly more crossings in the regions between
the dipoles, as highlighted by the magnification. In summary, the lobes
between the dipoles boost the transport of chaotic trajectories in the phase
space, due to the size and also the increase in the number of crossings of the
manifolds.

As is well known , the stable and unstable manifolds are non-attracting
invariant sets that play a fundamental role in the dynamics of the chaotic
systems [31]. The results of Figures 5 and 6 support this idea. Hence, we
study a non-attracting invariant set as a chaotic saddle. Let $\mathcal{R}$ be
a limited region in the phase space, in our case $\mathcal{R}$ is the domain
$\mathcal{D}:=\\{(x,y)|x\in[0,1]~{}\text{and}~{}y\in[-1,1]\\}$. There is a set
of points which remain in $\mathcal{R}$ for all time under forward iteration
of the ${SNM}$, and another set of points remain in the region for all time
under the backward iteration of the map. These sets are the stable and
unstable sets, respectively, and their union is the chaotic saddle [32]. We
compute an approximation of the chaotic saddle by the “sprinkler method”
proposed in [31]. The method consists of placing points on a fine grid over
the region $\mathcal{R}$ and iterating all the points until a final iteration
$n_{F}$. The initial conditions of the trajectories that stay in the region
during all the iterations approximate the stable set, their final positions
approximate the unstable manifold, and the iteration ${n_{F}}/{2}$
approximates the chaotic saddle.

To calculate the chaotic saddle for the phase space plots of Figure 5, we use
a grid of $2000\times 2000$ initial conditions in $\mathcal{D}$, discard
points that belong to the islands, and consider the final iteration
$n_{F}=100$. The chaotic saddle is show in Figure 7. From Figure 7 and the
value of the transmissivity for each of the cases of Figure 5, we can confirm
that there is a relationship between the spatial distribution of the invariant
set and the transport through the phase space. The saddle of Figure 7 (a) is
related to the case with the TFB: here the chaotic saddle is dense and
distributed in the whole domain of $x$. Visually, the density appears uniform
on the whole saddle. A different scenario is observed in Figure 7 (b). In this
case, we observe that the chaotic saddle is denser around the island of period
2 and the saddle has “holes” and “channels”, i.e., paths for the chaotic
orbits to cross the phase space with no restriction. Lastly, the chaotic
saddle for the odd scenario, Figure 7 (c), is dense around the islands but we
can also observe the channels inside the saddle. With these results, it seems
that a uniformly distributed and continuum saddle is related to a scenario
with no transport through the phase space, while the saddles that are not
uniformly distributed and are subdivided can lead to a scenario with non-null
transport.

Figure 7: Approximation of the chaotic saddle correspondent to the phase
spaces of Figure 5. The computation is made in a grid of $2000\times 2000$
initial conditions $n_{F}=100$. The black points represent the iteration
$n=50$ for orbits that do not escape the region
$\mathcal{D}:\\{(x,y)|x\in[0,1]~{}\text{and}~{}y\in[-1,1]\\}$ in 100
iterations.

##  6 Conclusions

The transport of chaotic trajectories across the phase space is prevented in
nontwist Hamiltonian systems by the existence of the shearless curve, an
invariant set that is robust to strong perturbations. After the shearless
curve breakup, the transport is affected by the structures formed by the
stable and unstable manifolds related to the twin chains of hyperbolic points.
We analyzed the transport in nontwist systems by using the paradigmatic
standard nontwist map, and we observed the parity of the twin islands plays a
crucial role in the collective motion of chaotic trajectories. For even period
islands the transport can be null, even though there is not a complete
barrier, while for the odd scenario we observe the emergence of transport
immediately after the shearless curve breakup.

Besides the transmissivity, the parity of the islands also affects the escape
basin and the relative escape times. For the odd case, we observe an intricate
boundary between the escape basins and larger escape times are related to the
stickiness and the hierarchy of the islands around islands structure. The
boundary between the escape basin is smooth for the even case, and the escape
times are related to the manifold structure that emerges in the even scenario
case.

We justified the difference between the transport for the even and odd cases
by examining the structures formed by the manifolds related to the chains of
hyperbolic points. For the odd case, we saw the well-known structure of
manifolds that can act like turnstiles for the chaotic trajectories. The main
novelty of the present paper is the structure formed by the manifolds in the
even scenario. In this case, we observed a dipole-like structure formed by the
manifolds of different chains of hyperbolic points, characterizing an
intercrossing scenario when the period of the islands is even.

For the even scenario, the intercrossing always occurs, but this does not mean
high transport, as in the odd case. The lobes formed in the intercrossing are
small, decreasing the possibility of transport. The low/high transport can
also be described by the chaotic saddle. The density and the continuity of the
chaotic saddle play important roles in the transport: dense and continuous
saddles decrease the transport, while subdivided saddles facilitate the
transport through phase space.

##  7 Acknowledgments

This research received the support of the National Council for Scientific and
Technological Development (CNPq - Grant No. 309670/2023-3, 403120/2021-7 and
301019/2019-3) and Fundação de Amparo à Pesquisa do Estado de São Paulo
(FAPESP) under Grant No. 2018/03211-6, 2022/12736-0, and 2023/10521-0. PJM was
supported by U.S. Dept. of Energy Contract $\\#$DE-FG05-80ET-53088.

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