High Density Nodes in the Chaotic Region of 1D Discrete Maps

We report on the definition and characteristics of nodes in the chaotic region of bifurcation diagrams in the case of 1D mono-parametrical and S-unimodal maps, using as guiding example the logistic map. We examine the arrangement of critical curves, the identification and arrangement of nodes, and the connection between the periodic windows and nodes in the chaotic zone. We finally present several characteristic features of nodes, which involve their convergence and entropy.


Critical Lines in the Chaotic Zone
As we observe in Figure 1a, the chaotic zone is characterized by Critical Curves (CC), which are the loci of enhanced density of points. Figure 3a-e shows the formation of the chaotic zone as the number of iterations increases. We observe the multi-folding of chaotic orbits in the chaotic zone, while the critical curves appear to be the locus of the extrema of the orbits due to their folding, as shown in Figure 3e. This is reasonable because the density of mapped points P(x; p) is, by construction, inversely proportional to the derivative of the map

Critical Lines in the Chaotic Zone
As we observe in Figure 1a, the chaotic zone is characterized by Critical Curves (CC), which are the loci of enhanced density of points. Figure 3a-e shows the formation of the chaotic zone as the number of iterations increases. We observe the multi-folding of chaotic orbits in the chaotic zone, while the critical curves appear to be the locus of the extrema of the orbits due to their folding, as shown in Figure 3e. This is reasonable because the density of mapped points P(x; p) is, by construction, inversely proportional to the derivative of the map ∂x t /∂p = ∂ f (t) (x 0 ; p)/∂p. The critical curves of multiplicity n = 0 and n = 1 correspond to those surrounding the chaotic zone at its upper and lower edge, respectively. The curve of multiplicity n = 2 is the dominant one located in the interior of SCB, passing through the band-merging Q1 and the end of the chaotic zone at the point (p = 4, x = 0). (In fact, all critical curves pass through the point (p = 4, x = 0), except the chaotic orbit of n = 0, which passes through the point (p = 4, x = 1)).
All the higher ranking critical curves (n ≥ 3) have maxima and minima located inside WOMs of periods equal to the multiplicity of the curves. It must be noted that these maxima or minima of the critical curves are located inside the miniature's zones of order (not exactly at the branches). However, the critical curves of the chaotic main bifurcation diagrams have similar role in the miniature chaotic zones, but with different multiplicity (Figure 4). For example, the curve of n = 3 has a single maximum; this is located inside the WOM of period 3. This WOM starts at p = 2√2 + 1 = 3.82842712474619, the double period bifurcation starts at p ≈ 3.8414 …, while the maximum is located at p ≈ 3.8390 …. The chaotic curve of multiplicity n = 3 in the main chaotic zone becomes of multiplicity n = 3:3 = 1 in the secondary chaotic zone (that is the critical curve passing through the lowest x-values). It is interesting that, in general, the product of the multiplicity of a critical curve with the period of the WOM is a constant. In the example of WOM-3, multiplicity is 3 when periodicity is 1 (main chaotic zone), while multiplicity is 1 when periodicity is 3, i.e., 3 × 1 = 1 × 3. Consequently, the critical curves, with multiplicity n in the main chaotic zone, do not appear in WOMs with period larger than n. The critical curves can be expressed by i.e., X(p; 0) = 1 4 p, X(p; 1) = 1 16 where n is the multiplicity of the critical curves. (The derivation of Equation (2a) is shown in Section 4). The critical curves of multiplicity n = 0 and n = 1 correspond to those surrounding the chaotic zone at its upper and lower edge, respectively. The curve of multiplicity n = 2 is the dominant one located in the interior of SCB, passing through the band-merging Q 1 and the end of the chaotic zone at the point (p = 4, x = 0). (In fact, all critical curves pass through the point (p = 4, x = 0), except the chaotic orbit of n = 0, which passes through the point (p = 4, x = 1)).
All the higher ranking critical curves (n ≥ 3) have maxima and minima located inside WOMs of periods equal to the multiplicity of the curves. It must be noted that these maxima or minima of the critical curves are located inside the miniature's zones of order (not exactly at the branches). However, the critical curves of the chaotic main bifurcation diagrams have similar role in the miniature chaotic zones, but with different multiplicity (Figure 4). For example, the curve of n = 3 has a single maximum; this is located inside the WOM of period 3. This WOM starts at p = 2 √ 2 + 1 = 3.82842712474619, the double period bifurcation starts at p ≈ 3.8414 . . . , while the maximum is located at p ≈ 3.8390 . . . . The chaotic curve of multiplicity n = 3 in the main chaotic zone becomes of multiplicity n = 3:3 = 1 in the secondary chaotic zone (that is the critical curve passing through the lowest x-values). It is interesting that, in general, the product of the multiplicity of a critical curve with the period of the WOM is a constant. In the example of WOM-3, multiplicity is 3 when periodicity is 1 (main chaotic zone), while multiplicity is 1 when periodicity is 3, i.e., 3 × 1 = 1 × 3. Consequently, the critical curves, with multiplicity n in the main chaotic zone, do not appear in WOMs with period larger than n.  Figure 3e, we plot the upper periodic attractor inside WOM of period 3 (and the produced secondary bifurcation diagram), and co-plot the critical curves with multiplicity n = 0 (blue) and n = 3 (green). We observe that the chaotic curves of the main chaotic zone appear also in miniature chaotic zones inside WOMs but with smaller multiplicity. Indeed, the critical curve with multiplicity n = 3 appears to pass through the lowest x-values of the miniature chaotic zone, that is, the role of the critical curve with multiplicity n = 1. What is happening is that if n is the multiplicity of a critical curve in the main chaotic zone, then in a WOM of period T, the same critical curve becomes of multiplicity n/T. If this ratio is less than 1, then the critical curve does not appear in that WOM.
Similarly, the critical curve of n = 4 has a single maximum located at the WOM of period 4, at p = 3.960101882689952. However, the curve of n = 5 has three maxima, each located to one of the three WOM of period 5 that exist in SCB. The critical curve of n = 6 also has three maxima at SCB, corresponding to the three WOMs of period 6 in SCB. However, there is one more maximum, which is located at the WOM of period 3 in the chaotic bands of n = 1, i.e., CB(0) and CB(1), where the chaotic orbits have period doubled than the respective ones in SCB, i.e., the chaotic orbits at the WOM of period 3 correspond to a total period 2 × 3. Figure 5 plots the chaotic zone, together with the critical curves modelled by Equation (2). We observe the curves of multiplicity from 0 to 6 in SCB. We also observe some other characteristic property of these curves, that is, their divergence from and convergence to points called nodes.   Figure 3e, we plot the upper periodic attractor inside WOM of period 3 (and the produced secondary bifurcation diagram), and co-plot the critical curves with multiplicity n = 0 (blue) and n = 3 (green). We observe that the chaotic curves of the main chaotic zone appear also in miniature chaotic zones inside WOMs but with smaller multiplicity. Indeed, the critical curve with multiplicity n = 3 appears to pass through the lowest x-values of the miniature chaotic zone, that is, the role of the critical curve with multiplicity n = 1. What is happening is that if n is the multiplicity of a critical curve in the main chaotic zone, then in a WOM of period T, the same critical curve becomes of multiplicity n/T. If this ratio is less than 1, then the critical curve does not appear in that WOM.
Similarly, the critical curve of n = 4 has a single maximum located at the WOM of period 4, at p = 3.960101882689952. However, the curve of n = 5 has three maxima, each located to one of the three WOM of period 5 that exist in SCB. The critical curve of n = 6 also has three maxima at SCB, corresponding to the three WOMs of period 6 in SCB. However, there is one more maximum, which is located at the WOM of period 3 in the chaotic bands of n = 1, i.e., CB(0) and CB(1), where the chaotic orbits have period doubled than the respective ones in SCB, i.e., the chaotic orbits at the WOM of period 3 correspond to a total period 2 × 3. Figure 5 plots the chaotic zone, together with the critical curves modelled by Equation (2). We observe the curves of multiplicity from 0 to 6 in SCB. We also observe some other characteristic property of these curves, that is, their divergence from and convergence to points called nodes.  , and co-plot the critical curves with multiplicity n = 0 (blue) and n = 3 (green). We observe that the chaotic curves of the main chaotic zone appear also in miniature chaotic zones inside WOMs but with smaller multiplicity. Indeed, the critical curve with multiplicity n = 3 appears to pass through the lowest x-values of the miniature chaotic zone, that is, the role of the critical curve with multiplicity n = 1. What is happening is that if n is the multiplicity of a critical curve in the main chaotic zone, then in a WOM of period T, the same critical curve becomes of multiplicity n/T. If this ratio is less than 1, then the critical curve does not appear in that WOM.
Similarly, the critical curve of n = 4 has a single maximum located at the WOM of period 4, at p = 3.960101882689952. However, the curve of n = 5 has three maxima, each located to one of the three WOM of period 5 that exist in SCB. The critical curve of n = 6 also has three maxima at SCB, corresponding to the three WOMs of period 6 in SCB. However, there is one more maximum, which is located at the WOM of period 3 in the chaotic bands of n = 1, i.e., CB(0) and CB(1), where the chaotic orbits have period doubled than the respective ones in SCB, i.e., the chaotic orbits at the WOM of period 3 correspond to a total period 2 × 3. Figure 5 plots the chaotic zone, together with the critical curves modelled by Equation (2). We observe the curves of multiplicity from 0 to 6 in SCB. We also observe some other characteristic property of these curves, that is, their divergence from and convergence to points called nodes.

Identification and Arrangement of Nodes in the Chaotic Zone
The density of the map values, x t , after t iterations of an initial value x 0 in the interval (0, 1), denoted by P(x; p), is given by (e.g., see: [10,22,33]): Next, we construct the density P(x; p) for nonlinear parameter values taken at the beginning of SCB. In Figure 6b-e, we start with p = 3.7, and then, decrease p until we reach Q 1 , located at p = 3.678573510428320. Each density peak indicates a critical curve, cut at the certain value of p. We observe that there are plenty of peaks, most of them undistinguished from each other, but as the nonlinear parameter approaches Q 1 , they are all accumulated into three main peaks; two of these correspond to the peaks at the upper and lower edges of SCB, while the third corresponds to a node, located at the point that connects CB(0), CB(1), and SCB. This node is shown in Figure 6a that magnifies the local region of chaotic zone. We observe that the node is a point of intersection of critical curves.
The node, identified in Figure 6e, corresponds to a density profile at p = Q 1 with exactly three peaks, two at the upper/lower edges, and one at the interior. For some larger value of the nonlinear parameter, p = 3.9277337001786751, there is a node corresponding to a density profile with four single peaks. As we will see further below, CBs have nodes that correspond to density profiles with any number of peaks; the nodes, the number of the peaks at their density profiles, and their arrangement in the CBs of the chaotic zone, are interwoven with the period and arrangement of WOMs.

Identification and Arrangement of Nodes in the Chaotic Zone
The density of the map values, xt, after t iterations of an initial value x0 in the interval (0, 1), denoted by P(x; p), is given by (e.g., see: [10,22,33]): Next, we construct the density P(x; p) for nonlinear parameter values taken at the beginning of SCB. In Figure 6b-e, we start with p = 3.7, and then, decrease p until we reach Q1, located at p = 3.678573510428320. Each density peak indicates a critical curve, cut at the certain value of p. We observe that there are plenty of peaks, most of them undistinguished from each other, but as the nonlinear parameter approaches Q1, they are all accumulated into three main peaks; two of these correspond to the peaks at the upper and lower edges of SCB, while the third corresponds to a node, located at the point that connects CB(0), CB(1), and SCB. This node is shown in Figure 6a that magnifies the local region of chaotic zone. We observe that the node is a point of intersection of critical curves.
The node, identified in Figure 6e, corresponds to a density profile at p = Q1 with exactly three peaks, two at the upper/lower edges, and one at the interior. For some larger value of the nonlinear parameter, p = 3.9277337001786751, there is a node corresponding to a density profile with four single peaks. As we will see further below, CBs have nodes that correspond to density profiles with any number of peaks; the nodes, the number of the peaks at their density profiles, and their arrangement in the CBs of the chaotic zone, are interwoven with the period and arrangement of WOMs. We rank the nodes as follows: The order of a node is given by the number of the peaks that surround the node in the corresponding density profile in the interior of the CB, namely, excluding the two edges and the node itself. Therefore, the node at p = Q1 ≡ N0 is of order 0, while the node at p = 3.9277337001786751 ≡ N1 is of order 1. Both nodes are shown in the SCB plotted in Figure 7. We rank the nodes as follows: The order of a node is given by the number of the peaks that surround the node in the corresponding density profile in the interior of the CB, namely, excluding the two edges and the node itself. Therefore, the node at p = Q 1 ≡ N 0 is of order 0, while the node at p = 3.9277337001786751 ≡ N 1 is of order 1. Both nodes are shown in the SCB plotted in Figure 7. As the nonlinear parameter p increases beyond the node of order 1, we find a node of order 2 at p = 3.982570733172925 ≡ N2. In general, there is only one node of order n + 1 at the right-hand side of the node of order n. Thus, the location of these nodes at the nonlinear parameter axis is arranged as follows: N0 < N1 < N2 < … < Nn < Nn+1 < … < N∞ = 4. These types of nodes are called primary. Table A1 in Appendix A contains the positions Nn of primary nodes, up to the order 8. Figure 8 plots the density profiles at the band-merging, the primary nodes up to the 4th order, and the case of fully developed chaos at p = 4, which corresponds to the limit where all the primary nodes converge and is symbolized as N∞.  As the nonlinear parameter p increases beyond the node of order 1, we find a node of order 2 at p = 3.982570733172925 ≡ N 2 . In general, there is only one node of order n + 1 at the right-hand side of the node of order n. Thus, the location of these nodes at the nonlinear parameter axis is arranged as follows: These types of nodes are called primary. Table A1 in Appendix A contains the positions N n of primary nodes, up to the order 8. Figure 8 plots the density profiles at the band-merging, the primary nodes up to the 4th order, and the case of fully developed chaos at p = 4, which corresponds to the limit where all the primary nodes converge and is symbolized as N ∞ . As the nonlinear parameter p increases beyond the node of order 1, we find a node of order 2 at p = 3.982570733172925 ≡ N2. In general, there is only one node of order n + 1 at the right-hand side of the node of order n. Thus, the location of these nodes at the nonlinear parameter axis is arranged as follows: N0 < N1 < N2 < … < Nn < Nn+1 < … < N∞ = 4. These types of nodes are called primary. Table A1 in Appendix A contains the positions Nn of primary nodes, up to the order 8. Figure 8 plots the density profiles at the band-merging, the primary nodes up to the 4th order, and the case of fully developed chaos at p = 4, which corresponds to the limit where all the primary nodes converge and is symbolized as N∞.  The arrows indicate the density peaks that correspond to critical curves (and not to the edges or the node itself), whose number defines the nodal order.

Mathematical Formulae of Critical Curves and Nodes
We construct the mathematical formulae that give the critical curves and the position of each primary node. We start by partitioning the map in a sequence of subintervals, { } ∞ =0 n n I , where In involves orbits with a specific number of n successive ascents before a descent occurs [16].
The fixed point given by ( ) 0 , respectively ( Figure 9). Let with an accumulation limit point at If is in (R), the orbit lies always below the diagonal, hence, only one descent is conceivable before the next sequence of ascents occurs, i.e., is in (L), the orbit lies always above the diagonal, hence, only subsequent ascents can occur. For n successive ascents, we have All the points of an orbit in are mapped after one ascent to subinterval (that is, the whole interval (R)), and then, from 0 I , they are mapped after a descent back to (L). In addition, all The arrows indicate the density peaks that correspond to critical curves (and not to the edges or the node itself), whose number defines the nodal order.

Mathematical Formulae of Critical Curves and Nodes
We construct the mathematical formulae that give the critical curves and the position of each primary node. We start by partitioning the map in a sequence of subintervals, {I n } ∞ n=0 , where I n involves orbits with a specific number of n successive ascents before a descent occurs [16].
The fixed point given by f (u 0 ) = u 0 separates the whole interval 0 ≤ x ≤ 1 in left-hand (L) and right-hand (R) sides, i.e., 0 ≤ x < u 0 and u 0 < x ≤ 1, respectively ( Figure 9). Let u n+1 be the preimage of u n , ∀n ≥ 0, defined by the reverse map, The arrows indicate the density peaks that correspond to critical curves (and not to the edges or the node itself), whose number defines the nodal order.

Mathematical Formulae of Critical Curves and Nodes
We construct the mathematical formulae that give the critical curves and the position of each primary node. We start by partitioning the map in a sequence of subintervals, , where In involves orbits with a specific number of n successive ascents before a descent occurs [16].
All the points of an orbit in are mapped after one ascent to subinterval (that is, the If x n = f (n) (x 0 ) is in (R), the orbit lies always below the diagonal, hence, only one descent is conceivable before the next sequence of ascents occurs, i.e., x n+1 = f (n+1) (x 0 ) < x n = f (n) (x 0 ). (Note: An ascent or a descent of an orbit is its jump to higher or lower x-values after one iteration.) If x n = f (n) (x 0 ) is in (L), the orbit lies always above the diagonal, hence, only subsequent ascents can occur. For n successive ascents, we have x < f (x) < f 2 (x) < · · · < f n (x). All the points of an orbit in subinterval I 1 = {x ∈ (u 1 , u 0 )} are mapped after one ascent to subinterval I 0 = {x ∈ (u 0 , 1)} (that is, the whole interval (R)), and then, from I 0 , they are mapped after a descent back to (L). In addition, all the points of an orbit in subinterval I 2 = {x ∈ (u 2 , u 1 )} are mapped after one ascent to subinterval I 1 = {x ∈ (u 1 , u 0 )}. In general, all the points of an orbit in subinterval I n = {x ∈ (u n , u n−1 )} are mapped after one ascent to subinterval I n−1 = {x ∈ (u n−1 , u n−2 )}, and so on, until I 0 is reached (Figure 9).
In this way, the map domain, 0 ≤ x ≤ 1, can be partitioned into subintervals I n , where n indicates the number of the successive ascents, before a descent occurs, namely: In the case of the logistic map, we find that is, for up to n = 3: The highest x-value of the orbit lies in subinterval I 0 for all the values of p in SCB (i.e., Q 1 < p < Q 0 = 4); at p = Q 1 , the highest x-value is x max = Q 1 /4, while at p = Q 0 = 4, the highest x-value becomes x max = 1. The maximum of the map is located at (x = 1/2, f (x) = p/4), which lies in subinterval I 1 ; thus, after one ascent, the maximum will be mapped to I 0 , reaching the highest possible x-value, given by Now, the lowest possible x-value is reached after a descent of the highest x-value, that is, mapped to Further iterations produce the critical curves of higher multiplicity (given in Equation (2)).
It is important to note that not all subintervals I n exist for a certain value of the nonlinear parameter p in SCB. As p increases, chaotic orbits reach higher and lower x-values. While the highest point will be always in subinterval I 0 , the lowest point can be in any subinterval I n , depending on the nonlinear parameter p. If the orbit is in subinterval I 1 , then ascents and descents interchange with each other in each iteration. In the chaotic zone, this happens only for p ≤ Q 1 , while there is only one point with this behavior and that is the band-merging p = Q 1 . Since there is just one point in SCB that belongs in subinterval I 1 , then this must be u 1 . Hence, the condition that applies in the case of band merging (or, primary node N 0 ) is: where we find The three peaks, shown in Figure 8a, represent the most frequent visits of the chaotic orbits for p = Q 1 (primary node N 0 ). Starting from the peak near the highest x-value, the orbits are mapped to the peak near the lowest x-value, and then, are mapped to the peak near the fixed point u 0 . The fact that there is only one ascent before reaching u 0 indicates that the lowest x-value is actually the preimage u 1 , verifying Equation (7a).
Similarly, the four peaks in Figure 8b represent the most frequent visits of the chaotic orbits at primary node N 1 . Orbits from near the highest x-value are mapped to the peak near the lowest x-value, then mapped to one more peak before are mapped to the peak near the fixed point u 0 . The three sequential ascents indicate that the lowest x-value is actually the preimage u 2 . Hence, the condition that applies in the case of the primary node N 01 is: where we find In the case of the primary node N n , we have Hence, we obtain Nonetheless, Equation (9a) is the one used for deriving the values of N n .
In Figure 10a, we plot the chaotic zone in SCB with the critical curves X(p; n) = f (n+1) 1 2 ; p (green) and preimages u n = f −1 L (n) (u 0 ) (blue), while in Figure 10b we show the intersections between x min (p) = f (2) ( 1 2 ; p) and u n = f −1 L (n) (u 0 ), for n = 0, 1, 2, 3. The intersections give the p-values of the primary nodes, which are plotted on a logarithmic scale in Figure 10c. We observe that the ratio of the p-values between two sequential nodes is constant and equal to 4. This is also shown in Figure 10d, where we plot 4 − N n as a function of n. We find that where the constant is A ≈ 0.06853892. In [16], we have shown that for p = 4 the preimages are given by u n+1 = sin 2 ( π 12 · 2 −n ) ≈ (π 2 /144) · 4 −n (cf., Equation (48c) in that paper), where (π 2 /144) ≈ 0.06853892; hence, it appears that the p-value of the primary nodes are connected with the preimages in the case of the fully-developed chaos (p = 4): where the constant is A ≈ 0.06853892. In [16], we have shown that for p = 4 the preimages are given by hence, it appears that the p-value of the primary nodes are connected with the preimages in the case of the fully-developed chaos (p = 4):

Connection between WOMs and Nodes
The primary nodes, defined by N0 < … < Nn < Nn+1 < … < N∞, are not the only ones appear in the chaotic zone. There is another more complicated configuration of secondary nodes. These nodes appear always in pairs surrounding WOMs. All the secondary nodes surrounding a WOM have the same order; the latter is also equal to the WOM's period.
The number of pairs of secondary nodes surrounding each WOM depends on the location with respect to the primary nodes. In the region between the primary nodes of order 0 and 1, called pyramid-1, the WOMs are surrounded by only one pair of secondary nodes; also, in the region between the primary nodes of order 1 and 2, called pyramid-2, the WOMs are surrounded by two pairs of secondary nodes, etc. For example, there is only one pair of secondary nodes surrounding the WOM of period three (Figure 11), but there are two pairs of secondary nodes surrounding the WOM of period four, etc. In general, the pyramid-n is defined by the region between the primary nodes of order n − 1 and n, the included WOMs are surrounded by n pairs of secondary nodes with order n, and the WOM of minimum period is the one with period n + 2, which is called main WOM of the pyramid. Therefore, we have the following equalities: Order of secondary nodes = Period of surrounded WOM (11a) and Number of secondary nodal pairs = Order of next primary node = Period of main WOM − 2 (11b) The two secondary nodes of each pair are located one at each side of the surrounded WOM; for

Connection between WOMs and Nodes
The primary nodes, defined by N 0 < . . . < N n < N n+1 < . . . < N ∞ , are not the only ones appear in the chaotic zone. There is another more complicated configuration of secondary nodes. These nodes appear always in pairs surrounding WOMs. All the secondary nodes surrounding a WOM have the same order; the latter is also equal to the WOM's period.
The number of pairs of secondary nodes surrounding each WOM depends on the location with respect to the primary nodes. In the region between the primary nodes of order 0 and 1, called pyramid-1, the WOMs are surrounded by only one pair of secondary nodes; also, in the region between the primary nodes of order 1 and 2, called pyramid-2, the WOMs are surrounded by two pairs of secondary nodes, etc. For example, there is only one pair of secondary nodes surrounding the WOM of period three (Figure 11), but there are two pairs of secondary nodes surrounding the WOM of period four, etc. In general, the pyramid-n is defined by the region between the primary nodes of order n − 1 and n, the included WOMs are surrounded by n pairs of secondary nodes with order n, and the WOM of minimum period is the one with period n + 2, which is called main WOM of the pyramid. Therefore, we have the following equalities: Order of secondary nodes = Period of surrounded WOM (11a) and Number of secondary nodal pairs = Order of next primary node = Period of main WOM − 2 (11b) The two secondary nodes of each pair are located one at each side of the surrounded WOM; for example, the main WOM in pyramid-3, that is, the WOM of period 5, is surrounded by 2 × 3 secondary nodes of order 5, where 3 of those are located at the left-hand side and 3 at the right-hand side of that WOM. Table A2 in Appendix A shows the secondary nodes up to the 6th order located within the main chaotic band, SCB. Figure 12 shows a sketch of SCB, where the primary and secondary nodes are shown.  In every CB, WOMs have a specific arrangement, universal for all the one-dimensional unimodal maps (due to a property called structural universality, [24,26,27,47]). Under each main WOM there is a pyramidal configuration of infinite number of WOMs: Every WOM is surrounded by infinite number of WOM pairs, whose period is larger than the parental WOM's period, at least by two  In every CB, WOMs have a specific arrangement, universal for all the one-dimensional unimodal maps (due to a property called structural universality, [24,26,27,47]). Under each main WOM there is a pyramidal configuration of infinite number of WOMs: Every WOM is surrounded by infinite number of WOM pairs, whose period is larger than the parental WOM's period, at least by two units. For example, the main WOM of period 3 is surrounded by pairs of WOMs of period 5, 7, 8, and so on, while period 6 does not exist and the same holds for any multiple of 3, i.e., the main WOM's period. Similarly, the main WOM of period 4 is surrounded by pairs of WOMs of period 6, 7, and so on, while WOMs of period 8, or of any other multiple of 4, do not exist. Table A3 in Appendix A gives the location of WOMs (nonlinear parameter at their starting point), in the main chaotic band (SCB) and for period up to 0. The sketch in Figure 13 describes the arrangement of WOMs in CBs. It is Figure 12. Sketch of the main chaotic band, SCB, of the chaotic zone. We show the location of the first four pyramids, their associated primary nodes, and the secondary nodes surrounding their main WOM. All the nodes, primary and secondary ones, are located at the orbit of period 1, x ∞ (p) = 1 − 1/p (dash line). The period of the main WOM (blue) is the same as the order of the surrounding secondary nodes. The number of the pairs of the surrounding secondary nodes is the same as the order of the next primary node (green).
In every CB, WOMs have a specific arrangement, universal for all the one-dimensional unimodal maps (due to a property called structural universality, [24,26,27,47]). Under each main WOM there is a pyramidal configuration of infinite number of WOMs: Every WOM is surrounded by infinite number of WOM pairs, whose period is larger than the parental WOM's period, at least by two units. For example, the main WOM of period 3 is surrounded by pairs of WOMs of period 5, 7, 8, and so on, while period 6 does not exist and the same holds for any multiple of 3, i.e., the main WOM's period. Similarly, the main WOM of period 4 is surrounded by pairs of WOMs of period 6, 7, and so on, while WOMs of period 8, or of any other multiple of 4, do not exist. Table A3 in Appendix A gives the location of WOMs (nonlinear parameter at their starting point), in the main chaotic band (SCB) and for period up to 0. The sketch in Figure 13 describes the arrangement of WOMs in CBs. It is interesting that the secondary nodes follow the same exact scheme as WOMs; the only difference is that nodes come in pairs surrounding each WOM.

Features of Nodes
The primary or secondary nodes can be the limit of one or more WOM sequences. Indeed, the locations (along the axis of the nonlinear parameter p) of the WOMs on the left-hand side of pyramid-n and of the WOMs on the right-hand side of pyramid-(n − 1) construct two sequences that both lead to the primary node n.
For example, the location of the primary node N0 (at the band-merging Q1) can be computed (aside from the mathematical formulation given in Section 4) by finding all the WOMs with odd periods starting from the WOM-3 along the left-hand side of the pyramid-1 (that is by decreasing the nonlinear parameter p). Alternatively, the same node can be computed by finding all the main WOMs of generation 1, i.e., along the chaotic bands CB(0) and CB(1), (that is by increasing the nonlinear parameter p). Both sequences are given in Table A4 in Appendix A, and plotted in Figure  14a (upper panel), where we observe that they both converge to the primary node p∞ → N0, while the exponential convergence of WOMs locations, Similarly, the primary node N1 can be computed by finding the WOMs with even periods starting from the WOM-4 along the left-hand side of the pyramid-2 (that is by decreasing p) and the WOMS with odd periods starting from the WOM-3 along the right-hand side of the pyramid-1 (that is by increasing p). The two sequences are given in Table A5 in Appendix A, and plotted in Figure 14b (upper panel), where we observe that they both converge to the primary node p∞ → N1, while the exponential convergence of WOMs locations, We also examine how the primary nodes Nn converge to the full-developed chaos at p = 4, which corresponds to the node N∞ (Table A1). The convergence is shown in Figure 14c (upper panel) with the exponential rate described by log A ≈ −0.5 and λ ≈ 0.61 for N∞ (lower panel). Figure 13. Sketch of the universal WOM-arrangement. Between any two consecutive primary nodes (green), WOMs are arranged in a pyramidal configuration: each WOM is surrounded by pairs of other WOMs with higher period, that is. The main WOMs (those with minimum period) are in the top of the pyramids (red).

Features of Nodes
The primary or secondary nodes can be the limit of one or more WOM sequences. Indeed, the locations (along the axis of the nonlinear parameter p) of the WOMs on the left-hand side of pyramid-n and of the WOMs on the right-hand side of pyramid-(n − 1) construct two sequences that both lead to the primary node n.
For example, the location of the primary node N 0 (at the band-merging Q 1 ) can be computed (aside from the mathematical formulation given in Section 4) by finding all the WOMs with odd periods starting from the WOM-3 along the left-hand side of the pyramid-1 (that is by decreasing the nonlinear parameter p). Alternatively, the same node can be computed by finding all the main WOMs of generation 1, i.e., along the chaotic bands CB(0) and CB(1), (that is by increasing the nonlinear parameter p). Both sequences are given in Table A4 in Appendix A, and plotted in Figure 14a (upper panel), where we observe that they both converge to the primary node p ∞ → N 0 , while the exponential convergence of WOMs locations, {p n } ∞ 1 , toward the node is shown in the lower panel. Similarly, the primary node N 1 can be computed by finding the WOMs with even periods starting from the WOM-4 along the left-hand side of the pyramid-2 (that is by decreasing p) and the WOMS with odd periods starting from the WOM-3 along the right-hand side of the pyramid-1 (that is by increasing p). The two sequences are given in Table A5 in Appendix A, and plotted in Figure 14b (upper panel), where we observe that they both converge to the primary node p ∞ → N 1 , while the exponential convergence of WOMs locations, {p n } ∞ 0 , toward the node is again shown in the lower panel. The latter is described by p n ∼ = p ∞ + A · 10 −λ·n (12) with log A ≈ −1 and λ ≈ 0.452 for N 0 , and log A ≈ −1.25 and λ ≈ 0.57 for N 1 . We also examine how the primary nodes N n converge to the full-developed chaos at p = 4, which corresponds to the node N ∞ (Table A1). The convergence is shown in Figure 14c (upper panel) with the exponential rate described by log A ≈ −0.5 and λ ≈ 0.61 for N ∞ (lower panel). The constant and its derivation was introduced by Feigenbaum [3][4][5]64] to describe the convergence of bifurcation points in the zone of order, but the same holds for the band-merging reverse cascade (e.g., [13]). In general, it can be applied to any convergence sequence in the chaotic zone. In Figure 15, we plot the computed Feigenbaum sequences   The Feigenbaum constant δ is defined as the limit of the following sequence {F n } ∞ 0 : The constant and its derivation was introduced by Feigenbaum [3][4][5]64] to describe the convergence of bifurcation points in the zone of order, but the same holds for the band-merging reverse cascade (e.g., [13]). In general, it can be applied to any convergence sequence in the chaotic zone. In Figure 15, we plot the computed Feigenbaum sequences {F n } ∞ 0 and their limits, the Feigenbaum constant δ, which correspond to the convergence sequences plotted in Figure 14. The convergence to δ = F ∞ is shown in the upper panels of Figure 15, while the exponential rate of the convergence is shown in the lower panels. The constant and its derivation was introduced by Feigenbaum [3][4][5]64] to describe the convergence of bifurcation points in the zone of order, but the same holds for the band-merging reverse cascade (e.g., [13]). In general, it can be applied to any convergence sequence in the chaotic zone. In Figure 15, we plot the computed Feigenbaum sequences   We find that the Feigenbaum constants corresponding to the sequences converging toward the primary nodes N 0 and N 1 are δ ≈ 2.817612 and δ ≈ 3.715, respectively. It is worth mentioning that the same constant characterizes any of the two sequences, which approach the nodes from the left or from the right. Moreover, we find that the convergence of the sequence of primary constants toward N ∞ is characterized by a peculiar Feigenbaum constant, that is, exactly δ = 4.
Finally, we examine the entropy near the nodes. Given the distribution density at p, P(x; p), the entropy in the continuous description is given by the standard Shannon's form [65]: where σ x is the smallest scale that characterizes the physical quantity x (e.g., see: [66,67]). The discretization of this interval is given by setting hence, the entropy is given by where [1/σ x ] denotes the closer integer to 1/σ x . We also use another entropic form, generalized according to the Tsallis mono-parametrical formalism [68,69], where recovers the Shannon's entropy in Equation (16) for q → 1. We find the entropy is~1 at the primary nodes for q~1.985. We chose this specific q-index for computing the entropy, as for smaller q-indices, the entropy increases abruptly, while for larger q-indices it tends to zero. We find that the entropy increases as the nonlinear parameter approaches the nodes. Figure 16 plots the entropy calculated for q = 2 and [1/σ x ] = 100, where we observe that it increases when p is taken closer to the primary nodes N 1 and N 2 .
Entropy 2018, 20, 24 15 of 21 We find that the Feigenbaum constants corresponding to the sequences converging toward the primary nodes N0 and N1 are δ ≈ 2.817612 and δ ≈ 3.715, respectively. It is worth mentioning that the same constant characterizes any of the two sequences, which approach the nodes from the left or from the right. Moreover, we find that the convergence of the sequence of primary constants toward N∞ is characterized by a peculiar Feigenbaum constant, that is, exactly δ = 4.
Finally, we examine the entropy near the nodes. Given the distribution density at p, ( ) p x; P , the entropy in the continuous description is given by the standard Shannon's form [65]: where x σ is the smallest scale that characterizes the physical quantity x (e.g., see: [66,67]).
The discretization of this interval is given by setting hence, the entropy is given by We also use another entropic form, generalized according to the Tsallis mono-parametrical formalism [68,69], where recovers the Shannon's entropy in Equation (16) for q → 1. We find the entropy is ~1 at the primary nodes for q ~ 1.985. We chose this specific q-index for computing the entropy, as for smaller q-indices, the entropy increases abruptly, while for larger q-indices it tends to zero. We find that the entropy increases as the nonlinear parameter approaches the nodes. Figure 16 plots the entropy calculated for q = 2 and [1/σx] = 100, where we observe that it increases when p is taken closer to the primary nodes N1 and N2.

Conclusions
The paper presented the arrangement and features of nodes in the chaotic zone of 1D unimodal and mono-parametrical discrete maps f(x), using as guiding example the logistic map. The nodes are high density and intersection points of the critical curves in the chaotic zone.

Conclusions
The paper presented the arrangement and features of nodes in the chaotic zone of 1D unimodal and mono-parametrical discrete maps f (x), using as guiding example the logistic map. The nodes are high density and intersection points of the critical curves in the chaotic zone. First, we examined the arrangement of critical curves in the chaotic zone, using the density of the variable x, and a simple empirical formula that describes these curves.
Second, we examined the identification and arrangement of nodes in the chaotic zone. The peaks of density profiles, taken across the nonlinear parameter axis, reveal the position of critical curves; as the nonlinear parameter approaches to a node, these peaks converge to each other, forming only a certain number of peaks that defines the order of the node. We found two types of nodes. (i) The primary nodes are defined by their unique sequential arrangement in the chaotic band: There is only one primary node of order n + 1 beyond the primary node of order n (along the nonlinear parameter axis). This arrangement separates chaotic bands in regions called pyramids. These are specific configurations of WOMs. (ii) The secondary nodes appear in pairs surrounding WOMs, where the order of nodes equals the period of the surrounded WOMs. The number of pairs is a characteristic of the pyramid.
Third, we examined the connection between WOMs and nodes. There is a universal arrangement of WOMs in chaotic bands, which can be divided into pyramidal configurations, separated by the primary nodes.
Finally, we examined several features of nodes, such as the convergence and entropy. We computed the characteristic convergence rates of the sequences of WOMs that converge into primary nodes, as well as the sequence of primary nodes that converge to the point of full-developed chaos. We computed the Feigenbaum constants related to these convergences, showing that each node has its characteristic Feigenbaum constant. The entropy analysis revealed that near the nodes the entropy has a local maximum value.
The following related science questions may be examined in future analyses: How do the arrangement and features of nodes vary for S-unimodal maps with a local maximum of differential order other than 2 (e.g., [13,31,58])? What is the mathematical formulation that describes the secondary nodes? Are there types of nodes other than the primary and secondary ones? Are there the same features of nodes into the 1D Poincare section return maps (e.g., [70,71])? How the nodes and their properties appear in higher dimensions (e.g., [72,73])? Acknowledgments: The work was supported by the project NNX17AB74G of NASA's HGI Program.

Conflicts of Interest:
The authors declare no conflict of interest.   3.678573510428329

Appendix A. Tables
Note: The sequence of main WOMs in CB(0) and CB(1) (CBs of generation n = 1) converges from the left to the primary node of order 0 (N 0 , that is, the band-merging Q 1 ). The sequence of WOMs on the left side of the pyramid-1 in SCB (CB of generation n = 0) converges from the right to the primary node of order 0 (N 0 ). Note: The sequence of main WOMs in CB(0) and CB(1) (CBs of generation n = 1) converges from the left to the primary node of order 1 (N 1 ). The sequence of WOMs on the left side of the pyramid-1 in SCB (CB of generation n = 0) converges from the right to the primary node N 1 .