Transport barriers with shearless attractors

We present a mechanism to generate a sequence of shearless curves or attractors to form a band of transport barriers. We consider the labyrinthic nontwist standard map to prepare a scenario with three shearless curves. Dissipation is introduced and three shearless attractors coexist, very close to each other. In both cases a collective transport barrier is formed.

Figure 1

The birth of shearless curves in the phase space as the parameter $b$ is increased. In panel (a) the red curve is the only shearless curve, which is associated with an extremum in the winding number, showed in panel (b); the green dots show the elliptic fixed points, and the purple stars and black arrows show the hyperbolic fixed points. In panel (c) there are two shearless curves and in panel (d) there are two extrema, while in panel (e) there are three shearless curves and in panel (f) there are three extrema. The parameters used were $a=0.1$, $\eta =3$, $r_1=0.2$, $r_2=-0.2$, and zero dissipation $\gamma =0$. In panels (a) and (b) $b=0.005$. In panels (c) and (d) $b=0.0051$. In panels (e) and (f) $b=0.006$.

Figure 2

Weak dissipation is introduced. In panel (a) $\gamma =1\times {10}^{-4}$ and the three shearless attractors are smoothly deformed in relation to the plot of Fig. 1. In panel (b) $\gamma =2\times {10}^{-4}$ and the central attractor is attracting the other two. In panel (c) $\gamma =3\times {10}^{-4}$ and the formation of a single attractor is accentuated. In panel (d) $\gamma =5\times {10}^{-4}$ and the shearless attractors collapse into only one attractor. The vertical lines are to guide the eyes.