Cascades of alternating smooth bifurcations and border collision bifurcations with singularity in a family of discontinuous linear-power maps

Figure 1.  2D bifurcation diagrams of $f$ in the $(a, S(b))$-parameter plane, where $S(b) = \arctan (b)$, for $\gamma = 0.5 $ in (a) and $\gamma = 1.5$ in (b); striped regions are related to coexistence, colored regions to attracting cycles of different periods, uncolored regions to attracting cycles of higher periods or chaotic attractors, grey regions to divergence. In (c) examples of map $f$ are shown

Figure 2.  Bifurcation diagrams $a$ vs $S(x) $ at $b = -5$, where $S(x) = \arctan (x)$. In (a) $\gamma = 0.5, $ in (b) $\gamma = 2$ and in (c) $\gamma = 1$, associated with the subcritical, supercritical and degenerate flip bifurcations of the 2-cycle $LR$, respectively

Figure 3.  Graph of map $g(x)$ given in (21). In (a): $a = -1.3$, $b = 1.5$ and $\gamma = 0.5<1;$ in (b) $a = -1.7$, $b = 10$ and $\gamma = 1.5>1$

Figure 4.  2D bifurcation diagrams of map $f$ in the $(a, b)$-parameter plane for $\gamma = 1.5$ in (a) and $\gamma = 3$ in (b)

Figure 5.  Maps $g(x)$ and $g^{2}(x)$ at $a = -1.7$, $b = 10$ and $\gamma = 1.5$. In the right panel the generic case starting a period doubling sequence is schematically shown

Figure 6.  Attracting 4-cycle $(ML)^{2}$ of map $g(x)$ corresponding to the 6-cycle $R^{2}LR^{2}L$ of map $f(x)$. The cycle is close to its border collision. In (b) the enlargement of the small rectangle indicated in (a). Here $a = -2.155$, $b = 10$, $\gamma = 1.5.$

Figure 7.  In (a), repelling 4-cycle $ML^{3}$ of map $g$ corresponding to the repelling 5-cycle $R^{2}L^{3}$ of map $f.$ In (b), the enlargement of the small rectangle marked in (a), which shows also the graph of $g^{8}(x)$ with an attracting 8-cycle $L^{2}(LM)^{3}$ of map $g$ corresponding to the attracting 11-cycle $L^{2}(LR^{2})^{3}$ of map $f.$ Here $a = -2.16$, $b = 10$, $\gamma = 1.5.$

Figure 8.  Qualitative representation of the S-fold and BCBs of map $g^{n}$ in a neighborhood of $x = 0, $ in all the possible cases, showing the shape of $g^{n}$ at the bifurcation and after: In (a) a fold-BCB; in (b) an S-fold, related to a local maximum of $g^{n}$ in $x = 0;$ in (c) a fold-BCB; in (d) an S-fold, related to a local minimum of $g^{n}$ in $x = 0.$

Figure 9.  In (a): fold-BCB leading to a pair of 3-cycles of map $g$; In (b): fold-BCB leading to a pair of 4-cycles of map $g$. Here $a = -2.5$, $b = 10$, $\gamma = 1.5.$

Figure 10.  1D bifurcation diagram as a function of $a$ at $b = 4$, $\gamma = 1.5$ for map $g$ in (a), and for map $f$ in (b). The values of $x$ are scaled as $y = \arctan (x)$ in order to show the values tending to $+\infty $

Figure 11.  Fold bifurcation of the pair of 2-cycles $LR$ at $a = -3$, $\gamma = 1.5 $ and $b = 0.35$ in (a), $b = 0.2$ in (b)

Figure 12.  1D bifurcation diagram of map $f$ as a function of $a$ at fixed $b = 1.9$ and $\gamma = 0.5.$ The variable $x$ is scaled by $y = \arctan (x)$. In (b) an enlargement of (a) is shown for $-1.1<a<-1$

Figure 13.  Graphs of map $g(x)$ at $\gamma = 0.5$ and $b = 0.5<b_{R}^{f}$. In (a) $a = -1.1$; in (b) $a = -2$; in (c) $a = -4.5.$

Figure 14.  Graphs of map $g$ at $\gamma = 0.5$ and $b = 2.2>b_{R}^{f}$. In (a) $a = -1.2$; in (b) $a = -1.6.$ The fixed point is the unique attractor