Set-oriented numerical computation of rotation sets

Figure 4.1.  Approximation of $\varrho{F_{1,1}}$ by a direct approach, $80$ (left) and $2500$ (right) iterations, grid range $0.001$ each

Figure 4.2.  Box image of a box $B$ in the box covering $\mathcal{B}_k$, for one test point, exemplarily

Figure 5.1.  Approximations $Q^*_{k,n}$ for the rotation set of the map $F_{1,1}$ with $k=8$ and $n = 1,2,5,10,25,50,100,200$ (from top left to bottom right)

Figure 5.2.  Zoom on top left area of the approximations $Q_{8,100}^\ast$ (left) and $Q_{8,200}^\ast$ (right) for the rotation set of the map $F_{1,1}$. The shaded area is the $2\sqrt{2}/n$-neighbourhood of these sets, which is a superset of $\rho(F_{1,1})$ by Lemma 19

Figure 5.3.  Approximations $Q_{k,n}^*$ for the rotation sets of the maps $F_{^1{/_2},^1{/_2}}, F_{1,{^1{/_4}}}, F_{^{3 }{/_{5}},^{3 }{/_{5}}}$ and $F_{{^3{/_4}},1}$ (with $k=50, 16, 50,45$ and $n=130,140,100,80$ from top left to bottom right)

Figure 5.4.  Approximation $Q_{60,130}^\ast$ for the rotation set of the map $G$

Figure 5.5.  Approximations $Q_{k,n}^*$ for the rotation sets of the perturbed maps $\bar{F}_{^1{/_2},^1{/_2}}, \bar{G}, \bar{F}_{1,{^1{/_4}}}, \bar{F}_{^{3 }{/_{5}},^{3 }{/_{5}}}$, $\bar{F}_{{^3{/_4}},1}$ and $\bar{F}_{1,1}$ (from top left to bottom right) according to Table 1

Figure 5.6.  Approximations $Q^*_{k,n}$ of the rotation set of $F_{0.873,0.873}$ with $n=50$ and $k=15,16,20,25,30,40,50,80$ (from top left to bottom right)

Figure 5.7.  Approximations $Q^*_{k,n}$ of the rotation set of the map $F_{0.873,0.873}$ with $n=100$ and $k=15,20,40,50$ (from top left to bottom right)

Figure 5.8.  Approximations of the rotation sets of $F_{\alpha_i,\beta_i}$ taken from a series with parameters $\alpha_i=\beta_i=0.02\cdot i$ and an adapted choice for the iteration time and grid sizes $n(i)=k(i)=110-i$. Pictures are shown for $i=30,35,36,37$ (from left to right), indicating a mode-locking region from $i=30$ to $i=35$

Figure 5.9.  A closer look at the parameter region considered in Figure 5.8. Rotation sets were approximated for parameters $\alpha_j=\beta_j=0.7+0.01\cdot j$, with $j=0, \ldots , 7$ and $n=75$ and $k=90$ fixed. Note the difference between the third picture in the first line and the third in Figure 5.8, which both correspond to parameters $\alpha=\beta=0.72$ (but different precisions)