Figure 1: Simply connected basin
D(
O) of the
fixed point O. In \(Z_0\) (below
LC ) a point has no preimage. In \(Z_2\) (above
LC) a point has two rank-one preimages represented by a letter with a lower index "-1" ("-2" for a rank-two preimage).
Example 1: Simply connected basin generated by a \(Z_{0}-Z_2\) map.
Figure 1 represents the phase plane of the map \(T\)\[x'=y,\] \(y'=0.8x+x^2+y^2\). The inverse map \(T^{-1}\) is given by\[ y=x',\] \(x=(-0.4\pm\sqrt{0.16+y'-x'^{2}})/2\). The critical curve \(LC\), \(y=x^{2}-0.16\) separates the plane into two open regions \(Z_0 \ (y-x^{2}+0.16<0)\) (no preimage) and \(Z_2 \ (y-x^{2}+0.16>0)\) (two distinct rank-one preimages). Here \(LC_{-1}=T^{-1}(LC)\), \(x=-0.4\) (the Jacobian of \(T\) vanishes), is the locus of two coincident rank-one preimages. It separates the regions \(R_1\) (sign "+" before the above square root), \(R_2\) (sign "-" before the above square root). The brown colored part is the basin \(D(O)\) of the attracting fixed point \(O\), \(\partial D(O) \) its boundary, which is the stable manifold \( W^{s}(P)\) of the saddle fixed point \(P\). The white region is the domain of diverging orbits. Due to the presence of the \(Z_0\) region:
\[
T[\partial D(O)\equiv W^{s}(P)] \subset \partial D(O)\equiv W^{s}(P),
\]
\[
T^{-1}[\partial D(O)\equiv W^{s}(P)] = \partial D(O)\equiv W^{s}(P)
\]
\(LC\) intersects the basin boundary at the points \(B\), and \(C\). Their rank-one preimages \(B_{-1}\), \(C_{-1}\), are located on \(LC_{-1}\). \(T^{-1}(B_{-1})=B_{-2}^{1}\cup B_{-2}^{2}\) gives the two rank-two preimages of \(B\), without preimage because located in \(Z_0\). Moreover \(T^{-1}(P)=P\cup P_{-1} \).
Example 2: bifurcations of basins and chaotic areas generated by a family of \(Z_{0}-Z_2\) maps
Figures 2, 3, 4, 5 represent the phase plane of the Kawakami's map \(T\) (\(Z_{0}-Z_2\) type)\[x'=ax+y, \ \ \ y'=b+x^2\].
This simple map is particularly rich in behaviors specific to
noninvertible maps, as all the bifurcations described below. For \(a=-0.42\), \(b=-1.09\) (Fig. 2), \(T\) has two
attractors at finite distance: a
period three chaotic area (grey colored) \((d_{1})\cup (d_{2})\cup (d_{3})\) (with the associated blue colored nonconnected basin), and an attracting invariant closed curve \(\Gamma\) (with the brown colored multiply connected basin). The white region is the domain of diverging orbits (basin of the attractor at infinity).
Figure 2:
a=-0.42,
b=-1.09\[P_{-2}^{2}\] belongs to \(Z_{0}\). The \(\Gamma\) multiply connected basin is brown colored. The non connected basin of the period three chaotic area (grey colored) \((d_{1})\cup (d_{2})\cup (d_{3})\) is blue colored. The white region is the domain of diverging orbits.
Figure 3:
a=-0.42,
b=-1.25. The square points (of period
k cycles,
k<20) inside the brown region (now basin of a period seven cycle) belong to a strange repeller, resulting from the destruction of the Fig. 2 chaotic area. The ones on the basin boundary belong to another strange repeller.
The critical curve \(L\) (\(y=b\)) separates a region \(Z_0\) from a region \(Z_2\). The locus of two coincident rank-one preimages is \(L_{-1}\) (\(x=0\)). The cusp point \(P \in \partial D\) is an unstable
node. Its rank-one preimage is \(P_{-1}\). The fixed point \(P\) has two rank-two preimages \(P_{-2}^{1}\) \(P_{-2}^{2}\). Note that \(P_{-2}^{2}\), which plays a fundamental role for the basin bifurcation, belongs to \(Z_{0}\) (a point has no preimage) in figs. 2, 3, and below Fig.4. In Fig. 5 it belongs to \(Z_{2}\) (a point has two distinct rank-one preimages).
For \(b=b_{c}\), \(-1.1182>b_{c}>-1.1183\), a contact bifurcation occurs between the boundary of the period three chaotic area \((d_{1})\cup (d_{2})\cup (d_{3})\) and the boundary of its immediate basin (the one containing the attractor). The period three chaotic area stops to be absorbing for this bifurcation parameter value.
When \(b=b_{c}-\varepsilon \), \(\varepsilon >0\) sufficiently small, the period three chaotic area is destroyed, ceasing to be an attractor. It turns into a
strange repeller (or
repellor) \(SR_{C}\), made up of infinitely many sequences of repelling cycles with increasing period, and their limit sets, constituting a
fractal unstable set.
Figure 4:
a=-0.42,
b=-1.35\[P_{-2}^{2}\] belongs to \(Z_{0}\). Multiply connected fractal basin (
D) obtained after the contact bifurcation of Fig. 3 in the master page (\(H_{0}\) is defined by this figure). The square period
k cycle points,
k<20, belong to a strange repeller, union of Fig. 2 ones. With its increasing rank preimages it constitutes the fractal limit set of the basin holes.
Figure 5:
a=-0.42,
b=-1.6. Simply connected basin of the chaotic area (
d), bounded by arcs \(L_r\) of critical curves of rank
r=0,1,...,6. Now \(P_{-2}^{2}\) belongs to \(Z_{2}\). The Fig. 4 holes (regions of diverging orbits) have been "opened" from the contact bifurcation\[P_{-2}^{2}\] located on the critical curve
L.
Figure 3 gives the situation when \(a=-0.42\), \(b=-1.25\). Now \(T\) has only one attractor, an attracting period seven cycle (represented by seven black stars) having \(D\) (the brown region) as basin. The colored points inside the brown region are those of period \(k\) unstable cycles, \(1<k<19\) (black for \(k>11\)). They belong to \(SR_{C}\). The points on the basin boundary belong to another strange repeller \(SR'_{C}\). The points \(SR_{C} \cup SR'_{C} \subset Z_2\) belong to the basin boundary \(\partial D\) which is fractal. An initial condition in the neighborhood of the \(SR_{C}\) points gives rise to a
chaotic transient before a regular convergence toward the period seven cycle. The parameter values \(a=-0.42\), \(b=-1.35\), give Fig. 4, the multiply connected basin of the period seven cycle, which corresponds to the qualitative Fig. 3c of the master page, after the
contact bifurcation of Fig. 3b of the master page (the symbols used are the same). The increasing rank preimages of \(H_0\) (holes belonging to the domain of diverging orbits) have \(SR_{C}\cup SR'_{C}\) as fractal limit set, which corresponds to one of the numerous cases of basin fractalization (cf. Mira
et al. [1996] and Bischi
et al. [2006]). Figure 5 (\(a=-0.42\), \(b=-1.6\) (now \(P_{-2}^{2}\) belongs to \(Z_{2}\)), is obtained after a
new contact bifurcation (\(b \simeq -1.415\)), which occurs when the point \(P_{-2}^{2}\in \partial D\) is located on the critical curve \(L\) (limit of existence of \(H_0\)). For \(a=-0.42\), \(b=-1.6\), the doubly connected red colored chaotic area \((d)\) is the only attractor at finite distance. Its basin \(D(d)\) (brown region) is now simply connected, the former holes having been "opened" on the former domain of diverging orbits. The basin boundary \(\partial D(d)\) is fractal (other fractalization type). The chaotic area \((d)\) is bounded by arcs of rank-\(h\) critical curves \(L_h\), \(h=0,1,2,...,6\), \(L_{0} \equiv L\).
Figure 6:
a=-0.42,
b=-1.64. Chaotic transient (red colored) toward infinity. It is obtained after a bifurcation due to the contact of the (
d) boundary with the boundary of its basin, which leads to the (
d) destruction. The strange repeller, generating this transient, belongs to the basin boundary of the attractor at infinity.
The chaotic area \((d)\) is destroyed, and turns into a strange repeller, when its boundary has a contact with the basin boundary \(\partial D(d)\) (\(b \simeq -1.63892\)). Figure 6 (\(a=-0.42\), \(b=-1.64\)) shows the corresponding situation of an orbit, which presents a chaotic transient (red colored) toward the attractor at infinity. It is obtained from the initial point (-0.95311327128; 0.57007252314), near a repelling period three cycle located in the former region \((d)\). With this initial point the orbit leaves the chaotic transient step after 16024 iterations (the chaotic transient duration depends on the initial point). Note that the iterated points with higher density are located in the neighborhood of critical curve arcs until the rank six. The brown points are points of the former basin (now destroyed), which remain inside the frame \(-50<x<50, -50<y<50\) during 28000 iterations. The strange repeller belongs to the basin boundary of the only map attractor, which is at infinity.
Examples related to other types of maps are given in Mira et al. [1996]. A case of \(Z_{1}-Z_{3}-Z_1\) map is dealt in Bischi et al. [2006].
