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SN Computer Science

, 1:26 | Cite as

Characterization of a New Potential Family of Organic-Like Pattern-Generating Dynamical Systems

  • David M. Marciel
Original Research
  • 310 Downloads

Abstract

In this paper, we characterize a new broad family of discrete-time dynamical systems, whose organic-like patterns resemble part of the external morphology of some families of invertebrates and the bioluminescence of some specific families of zooplankton. We also present a new type of fractal structure that is hidden inside the relationship between the main control parameters of this kind of systems. The approach to obtain these patterns lies in observing that they are visualized exclusively when the families of systems are interpreted as a whole, unique, global structure, emerging an organic-like pattern only when they are assembled. The characterization is based on three pillars: facts obtained from numerical computation, the analysis of the bifurcation diagrams, and the study of the new fractal structure. The aforementioned new broad family of discrete-time dynamical systems and a new type of fractal structure are formulated. The study of these new systems could help in finding unexpected alternatives to the representation of the patterns generated by the mechanisms of segmentation and bioluminescence, especially in invertebrates.

Keywords

Discrete-time dynamical systems Morphogenesis Bioluminescence Pattern formation Fractal 

Mathematics Subject Classification

37-02 37N25 92Bxx 37Exx 28A80 93C10 

Introduction

The study of dynamical systems and its different branches has evolved dramatically since Poincaré established the basis of their general theory in the nineteenth century. Subsequent progress has occurred in several fronts (bifurcation theory, specific models of behavior, low-dimensional systems and discrete-time dynamical systems, among others). Gurel and Rössler [1] wrote an accurate review of the development of the field up to the 70s of the twentieth century.

In September 1973, the international conference “Transformations ponctuelles et applications” was a pivotal point of reference for the study of nonlinear maps.1 It devoted a large part of its activity to two-dimensional recurrences and nonlinear maps and their applications. The conclusions of the conference are reflected in the work of Mira and Lagasse [2].

Gumowski and Mira [3] emphasized that the patterns generated by the dynamical systems must possess simultaneously a sufficient amount of complexity and regularity to be esthetically appealing.

Based on this general context, we characterize a new broad family of low-dimensional, nonlinear discrete-time dynamical systems, and describe their organic-like patterns. The patterns resemble part of the external morphology of some families of invertebrates.

There are former examples of organic-like pattern-generating systems, for instance,
  • The maps defined by Gumoswsky and Mira [3].

  • The classical reaction-diffusion models, for instance, compiled by Murray [4].

  • The mechanisms of segmentation (division of the body of some living beings into a series of repetitive segments) in biology. E.g., in 2012 it has been proved that there is a segmentation clock in insects, in a similar way to the vertebrate segmentation, which relies on a mechanism characterized by oscillating gene expression (Sarrazin et al. [5]). Besides, segmentation is studied as a self-organized patterning process, based on genetic oscillators. A recent study by J\(\ddot{\text {o}}\)rg et al. [6] reveals that the sequential pattern formation can be modeled by a set of signaling gradients, defining a framework to study the emergence of dynamic patterns. There is a mathematical formulation of the gradients, able to provide an appropriate model for the study of the patterns.

  • The mechanisms of bioluminescence. For instance, Herring [7] compiled a list of the genera of living organisms known or believed to contain luminous species.

Both the behavior and formulation of those models and their resulting patterns are very different from the ones we propose.

Our organic-like patterns could contribute as an alternative path to the simulation of the morphology of some families of invertebrates. In the patterns generated by our families of dynamical systems it is possible to observe regions that resemble the mechanisms of segmentation of living beings, and the accumulation of bioluminescence of some species. Additionally, following the aforementioned definition by Gumowski and Mira, the patterns obtained by our new method could be considered esthetically appealing.

The approach to obtain these new type of pattern-generating systems lies in observing that the patterns are visualized exclusively when the families of attractors are reviewed as a whole, unique, global structure. Each one of the maps of the family, in other words, each standalone bifurcation, does not present individually any remarkable visual property, apart from some basic rotations and symmetries, but together work as the pieces of a jigsaw, emerging an organic-like pattern only when they are assembled.

These new systems were found while studying2 diverse classical discrete-time families of attractors (especially the logistic map3, and the patterns obtained from the Chiritov–Taylor and Ikeda maps), reviewing the population models associated with some families of insects by May and Oster [11] and looking for new models4 on a trial and error basis.

Remark 1

The present characterization and results have been reached by direct numerical computation in all cases. Although the new patterns that we have observed resemble the segmentation and the accumulation of bioluminescence of some invertebrate families, initially there is no relationship between the mathematical expressions governing the segmentation mechanisms obtained from the studies on the matter and the mathematical expression of our families of systems. Thus, we believe that the study of these new patterns could help in finding unexpected alternatives to the representation of the aforementioned patterns, especially in invertebrates.

The main contents of the paper are as follows:
  • The next section shows the formulation of a new abstract family of dynamical systems, and one specific subfamily whose attractors are the target for the present article.

  • The following section provides the bifurcation diagrams of the targeted subfamily, and conclusions about the different types of dynamical systems (some of them periodic, others chaotic) that are associated with each bifurcation. We will explain the challenges found, due to the special nonlinearity of the family, to select an appropriate bifurcation diagram and study the behavior of the systems. It will also be described a new kind of fractal structure found hidden inside the relationship between the main control parameters.

  • The subsequent section describes some visual characteristics of the family of systems, specifically the top-level patterns, global symmetries and local asymmetries, and rotations and specular versions of the initial model.

  • Before the final section, the pattern-generating properties of these systems and comparison of the organic-like structures obtained from them with the external morphological patterns of some invertebrates at different levels of magnitude are shown. Supplementary online material is also provided.

  • Finally, a summary of the results and a list of open points regarding the study of the organic patterns and the taxonomy of the new fractal structure are presented.

Formulation of a New Family of Nonlinear Dynamical Systems

Fig. 1

Example of cyclic attractor. The map \(S\left( \frac{5102}{8 \times 10^3} 2\pi \right)\), for the initial conditions \((x_0=0,y_0=0)\), is spread along the global pattern of S(D) (white) into different independent clusters of points (green), each one of them containing points belonging to its periodic cycle (red) in their centers. The yellow square (below) is a partial zoom-in of the phase space of the map, including its transient regime (located in the green areas) and its periodic attractor (period 242, located in the red areas). The figure is visualizing the region \(z=a+bi\) / \(a,b \in [-5,5]\), where the complete family of cyclic attractors of the D-bifurcations S(D) is located

Fig. 2

Example of a non-periodic system. The map \(S\left( \frac{1068}{8 \times 10^3} 2\pi \right)\) is shown in green color. It is spread along the global pattern, generating several independent chaotic absorbing areas. The figure is visualizing the region \(z=a+bi\) / \(a,b \in [-5,5]\), where the complete family of attractors of the D-bifurcations S(D) is located

Fig. 3

Top–down: a Region \(z=a+bi\) / \(a,b \in [-5,5]\), where the complete family of chaotic absorbing areas of the D-bifurcations S(D) is located. b Same region, only showing the cyclic attractors. c The complete family of D-bifurcations, both the chaotic absorbing areas a and the cyclic attractors b

Definition 1

A broad family of new discrete-time dynamical systems is obtained from the following abstract dynamical system:
$$\begin{aligned} A(\alpha ,t,g) & = \{(x,y): (x_{n+1},y_{n+1}) \nonumber \\ & = (\mathfrak {I}(g(x_{n} + y_n i,t))\cdot \sin {\alpha }, \mathfrak {R}(g(x_n + y_n i, t))\cdot \cos {\alpha })\}, \end{aligned}$$
(1)
where
  • \(n \in \mathbb N\) is the iterator.

  • \(z_n=x_{n} + y_n i \in \mathbb C\) is a complex number and \(g(z,t): \mathbb {C}\) x \(\mathbb {R} \rightarrow \mathbb {C}\) is a complex function applied on each iteration n.

  • \(\mathfrak {I}(z): \mathbb {C} \rightarrow \mathbb {R}\) and \(\mathfrak {R}(z): \mathbb {C} \rightarrow \mathbb {R}\) are, respectively, the imaginary and real parts of the complex number obtained from g(zt).

  • The control parameters are defined as “time” \(t \in \mathbb R\), and “angle” \(\alpha \in [0,2\pi ) \in \mathbb R\). The use of t as a control parameter provides a driver to observe the instationary (time dependent) cinematic evolution of the patterns.

Definition 2

The abstract discrete-time version of the family (1) is expressed as follows:
$$\begin{aligned} B(D,t,g) & = {} \{(x,y): (x_{n+1},y_{n+1}) \nonumber \\ & = (\mathfrak {I}(g(x_{n} + y_n i,t))\cdot \sin {D}, \mathfrak {R}(g(x_n + y_n i, t))\cdot \cos {D})\}, \end{aligned}$$
(2)
where \(t>0 \in \mathbb {Q}\), and D belongs to the discretization:
$$\begin{aligned} D \in \left\{ 0, \frac{1}{{\text {Max}}D} 2\pi , \frac{2}{{\text {Max}}D} 2\pi , \ldots , \frac{{\text {Max}}D-1}{{\text {Max}}D} 2\pi \right\} \in [0,2\pi ). \end{aligned}$$
In other words, the control parameter \(\alpha\) has been discretized into a finite set of \({\text {Max}}D\) equidistant values along \([0,2\pi )\). We have computationally verified that \({\text {Max}}D=8 \times 10^3\) is a good enough maximum value to observe the behavior of the discrete-time family of dynamical systems.
The discrete model has the characteristics and follows the guidelines proposed by Sbalzarini in Appel and Feytmans’ book “Bioinformatics: A Swiss perspective” [13]:
  • It is delineated by complex shapes.

  • It is nonlinear.

  • It is plastic over time (time-varying dynamics).

  • The degrees of freedom of the system (control parameters D and t) involve timescales of similar order.

  • Using the same concepts exposed by Sbalzarini, whereas the abstract version of the family of systems (1) can be considered as an example of “relevant dynamics”, and could be modeled as differential equations, the complexity of the resulting system makes difficult to study its behavior. For that reason, we define a discrete version (2) to study the systems. This is considered the “slow dynamics” version of the family of dynamical systems.

  • The discretization of D and t reduces the problem to computing an approximation to the solution at a finite number of discretization points in space and for a finite number of time steps.

Definition 3

Our study will be focused on the subfamily \(g(z,t)=\frac{1}{(z+1)^{t}}\):
$$\begin{aligned} C(D,t)&= {} B\left( D,t,\frac{1}{(z+1)^{t}}\right) =\left\{\vphantom{\left( {\frac{1}{{((x_{n} + y_{n} i) + 1)^{t} }}} \right)} {(x,y):(x_{{n + 1}} ,y_{{n + 1}} )} \right. \\ & = \left( {\Im \left( {\frac{1}{{((x_{n} + y_{n} i) + 1)^{t} }}} \right)\cdot\sin D,} \right. \\ & \quad \left. {\left. {\Re \left( {\frac{1}{{((x_{n} + y_{n} i) + 1)^{t} }}} \right)\cdot\cos D} \right)} \right\}, \end{aligned}$$
(3)
where
  • based on the numerical results of the computational tests, iterating n up to \(8 \times 10^3\) is enough to study the behavior of the model and verify the existing organic-like patterns;

  • also based on the numerical results of the computational tests and their visualization, it is verified that the attractors, after removing the transient regime, finally reside in the region \(z=a+bi\) where \(a,b \in [-5,5]\).

Definition 4

The following subfamily of maps will be used to describe the different types of patterns emerging from this kind of systems:
$$\begin{aligned} S(D) = C(D,2) & = \left\{\vphantom{\left( {\frac{1}{{((x_{n} + y_{n} i) + 1)^{2} }}} \right)} {(x,y):(x_{{n + 1}} ,y_{{n + 1}} )} \right. \\ & = \left( {\Im \left( {\frac{1}{{((x_{n} + y_{n} i) + 1)^{2} }}} \right)\cdot\sin D,} \right. \\ & \quad \left. {\left. {\Re \left( {\frac{1}{{((x_{n} + y_{n} i) + 1)^{2} }}} \right)\cdot\cos D} \right)} \right\}. \\ \end{aligned}$$
(4)

The initial condition will henceforth be assumed to be \((x_0,y_0)=(0,0)\) to study the aforementioned subfamilies.

Remark 2

Chronologically, first we found the subfamily (4) on a trial and error basis (while studying, as mentioned in “Introduction”, diverse classical discrete-time families of attractors). Subsequently, to understand and properly categorize its nature and find other possible branches or subfamilies showing similar visual properties, the generic abstract version (1) and its progressively specialized expressions (2) and (3) were defined.

Since these systems were found on a trial and error basis, there is a fundamental question about their structural sensitivity (or, in other words, structural stability). Why these combinations of control parameters provide these kinds of organic-like patterns, and not other combinations? Do small changes in those control parameters (or, for instance in our case, a slight modification of the function g) imply new and quite different patterns? Our reference for this question is Adamson and Morozov’s study [14] about the structural sensitivity of dynamical systems. They proposed a method to explore multicomponent ecological models and demonstrate the existence of structural sensitivity. We totally agree with Adamson and Morozov’s conclusion: the structural sensitivity is an intrinsic property of biological models.

Thus, when constructing new models based on experimental trial and error basis, it is important to determine whether or not small changes in the mathematical expressions of those models can result in substantial changes in the results.

In our case, numerical computation5 shows that there is no sensitive dependence on the initial conditions as long as the values of the control parameters D and t are specified: the behavior of the family of systems will be similar independently of the initial condition \((x_0,y_0)\). Most part of the maps will reach nonchaotic cyclic attractors (a wide variety of cycles with different periods), before \(8 \times 10^3\) iterations; and the remaining cases will arrive to chaotic basins of attraction.

More than 93\(\%\) of the D-bifurcations of the family S(D) reach quickly their respective periodic cyclic attractors (e.g., Fig. 1).6 On the other hand, the remaining maps are falling into a chaotic regime (e.g., Fig. 2). For example, the first value of D not reaching a cyclic attractor before \(8 \times 10^3\) iterations is \(D=\frac{980}{8 \times 10^3} 2\pi\).

Regarding the maps falling into a chaotic regime, we believe (later this idea will be supported by the results of the bifurcation diagrams) that they will eventually reach a cyclic attractor after a very long iterative process. Initially, as we did not find computationally a periodic cycle for those remaining systems, these regions, at the current level of detail and for the purpose of our study, are considered chaotic absorbing areas.

When the complete subfamily (hence including every D-bifurcation) is visualized, spirals, clusters of points and other more elaborate shapes emerge, and, as a result of the combination of all of them, an organic-like pattern arises (Fig. 3).

Whereas a single D-attractor does not have a very clear visual pattern, the pattern and symmetries of the D-bifurcations, seen as a whole structure, can be easily distinguished. At a first sight, in a global scale, the x-axis divides the pattern into two specular halves. There are also replicated and rescaled subpatterns on both sides. Later we show that indeed there are local asymmetries: patterns that look similar but are not exactly equal in both sides. This characteristic (symmetry-breaking) resembles the symmetrical–asymmetrical patterns of the external morphology of living beings.

Summarizing, the family of D-bifurcations S(D) (4) is nonlinear, noninvertible, and depending on the cases might be chaotic (at least falling into long-term chaotic absorbing areas) and, in most of the cases, nonchaotic. These characteristics are also applicable to the subfamily C(Dt) (3). As a consequence of being time dependent (t), the subfamily of systems (3) is non-autonomous: a fixed set of initial conditions will not produce the same results for different values of the control parameter t.

Analysis of the Bifurcation Diagrams: Fractal Relationship Between the Control Parameters

Currently, we are analyzing the following subfamily:
$$\begin{aligned} S(D)&= {} \left\{\vphantom{\left( {\frac{1}{{((x_{n} + y_{n} i) + 1)^{2} }}} \right)} {(x,y):(x_{{n + 1}} ,y_{{n + 1}} )} \right. \\ & = \left( {\Im \left( {\frac{1}{{((x_{n} + y_{n} i) + 1)^{2} }}} \right)\cdot\sin D,} \right. \\ & \quad \left. {\left. {\Re \left( {\frac{1}{{((x_{n} + y_{n} i) + 1)^{2} }}} \right)\cdot\cos D} \right)} \right\} (4). \end{aligned}$$
There are three important points to understand the challenge to obtain a suitable bifurcation diagram:
  • The imaginary and real parts of the complex number \((x_{n},y_{n})\) are obtained from applying g(zt) to the previous iteration \((x_{n-1},y_{n-1})\) and then interchanging the real and imaginary parts of the resulting complex number (the imaginary part will become the real part of the next iteration, and the real part will become the imaginary part). Due to this characteristic, the resulting final expression is non-continuous, non-smooth and nonlinear. Indeed, there are other factors of nonlinearity, e.g., g(zt) is nonlinear too.

  • The control parameter D is applied to a sine and cosine functions, and a constant value of 2 has been assigned to t. Hence, the critical issue is the study of the bifurcation diagram of X and Y as a function of D.

  • The study of the bifurcations of (XY) as a function of D requires a three-dimensional graph, which is difficult to visualize. Thus, it is better to observe independently X as a function of D, and Y as a function of D, and then compare those results to understand the overall relationship.

Fig. 4

Bifurcation diagrams (DX) and (DY), \(D \in [0,2\pi )\) of the family S(D) for the initial conditions \((x_0=0, y_0=0)\). Clearly there are values of D in which the cyclic attractors are small, others in which the complexity of the cycle grows and finally, cases (green color) in which a cyclic attractor is not found computationally, thus leading to several chaotic absorbing areas. Due to the fact that D is a discrete variable whose values are equidistributed along \([0,2\pi )\), the diagrams are cylindrical (each diagram can join its left and right sides together)

Fig. 5

Visualization of the cyclic attractors of the subfamily S(D) for three different initial conditions. Left to right: \((x_0=0,y_0=0)\), \((x_0=-9102,y_0=828819.24313)\), and \((x_0=18920.12123,y_0=-121.0000012223)\). The accurate location, the shape and the density of the clusters of points differ slightly for each initial condition (the lighter they are, the denser they are), e.g., the clusters of points in the region inside the yellow squares have slightly different densities and shapes for each initial condition. The figure is visualizing a zoom-in of the region \(z=a+bi\) / \(a,b \in [-5,5]\), where the complete family of attractors of the D-bifurcations S(D) is located

Fig. 6

Fractal escape time map of \(C(D \in [0,2\pi ],t \in [0,10])\), for \((x_0=0,y_0=0)\). Whereas the y-axis is bounded by \([0,2\pi ]\), the x-axis, representing the evolution of time (t), is unbounded to the right. The bottom of the map continues naturally again at the top: it is a cylindrical fractal map

Fig. 7

(Top–down, left–right) Fractal escape time maps of \(C(D \in [\frac{400}{1000} 2 \pi , \frac{650}{1000}2 \pi ],t \in [3.00,5.50])\), \(C(D \in [\frac{590}{1000}2 \pi , \frac{615}{1000}2 \pi ],t \in [3.90,4.15])\), \(C(D \in [\frac{900}{1000}2 \pi , \frac{950}{1000}2 \pi ],t \in [5.00,5.50])\) and \(C(D \in [\frac{929}{1000}2 \pi , \frac{937}{1000}2 \pi ],t \in [5.08,5.16])\) for \((x_0=0,y_0=0)\)

Fig. 8

Escape time maps of S(17) (left) and C(2.5132741228718345, 3) (right) for the region of initial conditions \((x_0=[-600,600],y_0=[-600,600]), x_0,y_0 \in \mathbb Z\). In the second map, C(2.5132741228718345, 3), almost every initial condition is reaching a periodic cyclic attractor and the emerging pattern is a triskelion

Figure 4 shows separately the bifurcation diagrams of X and Y as a function of D. There are clearly values of D where the cyclic attractors are small, others where the complexity of the cycle grows and finally, areas where a cyclic attractor was not found computationally, which are the closest to the areas of the bifurcation diagrams where the longest cyclic attractors are.

In essence, by looking at these results, we deduce that the iterations required to reach a periodic cycle (if it exists) for the remaining so far unstable systems grows quickly, so it might happen that in some of the remaining cases the region is really a chaotic attractor, and not merely a temporary chaotic absorbing area or a periodic cyclic attractor.

When the initial condition \((x_0,y_0)\) is chosen at random, its orbit displays two distinct phases: an initial transient regime, and an asymptotic regime that persists for arbitrarily long iterations and is quantitatively and qualitatively equivalent to different D-bifurcations, as long as their D values are close to each other, e.g., the cyclic attractors (not including the chaotic absorbing areas) of three totally different initial conditions are visualized in Fig. 5, all of them leading to very similar qualitative and quantitative results. The families of attractors are visually very similar, but not exactly the same ones: there are subtle changes along the subpatterns.

As we stated at the beginning of the previous section, D is an equidistributed discretization, based on rational numbers (not necessarily natural numbers), of the domain of the control parameter \(\alpha \in [0,2\pi ) \in \mathbb R\) associated with the abstract family of dynamical systems \(A(\alpha ,t,g)\) (1). It is expressed as B(Dtg) (2).

The possibility of defining D as a rational number led us to study the relationship between D and t via an escape time algorithm.

Definition 5

The escape time algorithm of the subfamily of dynamical systems B(Dtg) (2) will evaluate, for a given initial condition \((x_0,y_0)\), how many iterations of n are required to reach a periodic cyclic attractor in less than a maximum value of \({\text {Max}}N\) iterations.7 If the initial condition requires more than \({\text {Max}}N\) iterations to attain a periodic cycle, it will be considered a chaotic absorbing area.8

The visualization of the algorithm is shown, for some selected intervals of D and t, in Figs. 6 and 7. The specific number of iterations required by the initial condition \((x_0=0,y_0=0)\) and a given combination of D and t to reach stability is represented by a color code. The regions in black color are not able to reach a periodic cyclic attractor before consuming \({\text {Max}}N\) iterations, so they are considered chaotic absorbing areas. The parameter t is located in the x-axis, and the parameter D is located in the y-axis.

Figure 6 shows the whole map, where \(D=0\) is located at the top and \(D=2\pi\) is at the bottom. Indeed, the bottom continues naturally again at the top: it is a cylindrical fractal map. The four figures included in Fig. 7 are zoomed regions of the whole map shown in Fig. 6. As a result, it is possible to visualize the basins of attraction for a specific initial condition \((x_0,y_0)\) and different combinations of the control parameters.

The relationship between D and t is of fractal nature. There is replication, rotation and resizing inside the patterns of the escape time map. It does not matter how much it is zoomed into the subregions, there will be areas in which the combination of D and t seems to provide a basin of attraction of \((x_0,y_0)\), such that the periodic cyclic attractor is always reached, and in other cases there are some regions of the relationship of D and t in which the initial condition will never reach a periodic cyclic attractor. Between these two types of subregions there are other unstable frontier regions in which a slight change makes the initial condition reach a periodic cycle or not. Thus, the relationship between D and t seems to be genuinely chaotic.

Remark 3

As far as we have been able to verify, there are no former examples of the study and visualization of these kind of fractal structures. Driven by these results, we hypothesize that the above escape time algorithm, applied to other classical dynamical systems, should lead to other type of hidden fractal structures (intrinsic to the peculiarities of each specific system).9

We have also visualized the escape time maps of the initial conditions of the region \((x_0=[-600,600],y_0=[-600,600]), x_0,y_0 \in \mathbb Z\) for fixed values of D and t, calculating the iterations of n required to reach a periodic cyclic attractor in less than a maximum value of \({\text {Max}}N\) iterations. In this case, the \(x_0\) values are located in the x-axis and the \(y_0\) values are located in the y-axis. Two examples are shown in Fig. 8.

As in the previous escape time test, if the initial condition requires more than \({\text {Max}}N\) iterations to attain a periodic cyclic attractor, it will be considered a chaotic absorbing area. The specific number of iterations required by the initial condition and a given combination of D and t to reach stability is represented by a color code. The regions in red color are not able to reach a periodic cyclic attractor before consuming \({\text {Max}}N\) iterations, so they are considered chaotic absorbing areas. A symmetrical pattern emerges in both examples.

Visual Characteristics, Symmetry–Asymmetry and Modifications of the Model

Fig. 9

Segmentation of the family S(D). The figure is visualizing the region \(z=a+bi\) / \(a,b \in [-5,5]\), where the complete family of attractors is located. The spatial regions are divided into different structures: head (red), body (green), antennae (blue) and limbs (brown)

Fig. 10

Modification of the signs of both the \(\mathfrak {I}\) and \(\mathfrak {R}\) values (top-left) of the family S(D), generating its three simple specular version: \(+\mathfrak {I}\) and \(-\mathfrak {R}\) (top-right, specular reflection on the x-axis), \(-\mathfrak {I}\) and \(+\mathfrak {R}\) (bottom-left, specular reflection on the y-axis) and \(-\mathfrak {I}\) and \(-\mathfrak {R}\) (bottom-right, specular reflection on the x- and y-axes). On each case, a slight variation of the densities (number of points accumulated in a specific region of the attractor) and the locations of the subpatterns happens, but the global shape does not change its general aspect. The figure is visualizing the region \(z=a+bi\) / \(a,b \in [-5,5]\), where the complete family of attractors is located

Fig. 11

Modification of the signs of the formulation of the family S(D) such that the points are relocated to their specular reflection only when their real part is positive. The figure is visualizing the region \(z=a+bi\) / \(a,b \in [-5,5]\), where the complete family of attractors is located

Fig. 12

Modification of the signs of the formulation of the family S(D) such that the points are relocated at their specular reflection depending on the quadrant where the points are initially located. The figure is visualizing the region \(z=a+bi\) / \(a,b \in [-5,5]\), where the complete family of attractors is located

Fig. 13

Modification of the constant coefficient of the polynomial \(z+1\) included in the denominator of \(g(z,t)=\frac{1}{(z+1)^{t}}\) of the family S(D), e.g., Y(D). The figure is visualizing the region \(z=a+bi\) / \(a,b \in [-5,5]\), where the complete family of attractors is located

We will now describe the main visual characteristics of the system, specifically the top-level patterns, the global symmetries and local asymmetries, and some manipulations of the formulation of the subfamily of D-attractors:
$$\begin{aligned} S(D)&= {} \left\{\vphantom{\left( {\frac{1}{{((x_{n} + y_{n} i) + 1)^{2} }}} \right)} {(x,y):(x_{{n + 1}} ,y_{{n + 1}} )} \right. \\ & = \left( {\Im \left( {\frac{1}{{((x_{n} + y_{n} i) + 1)^{2} }}} \right)\cdot\sin D,} \right. \\ & \quad \left. {\left. {\Re \left( {\frac{1}{{((x_{n} + y_{n} i) + 1)^{2} }}} \right)\cdot\cos D} \right)} \right\} (4). \end{aligned}$$
(5)

Definition 6

The proposed segmentation of the spatial region where the subfamily of dynamical systems (4) resides (Fig. 9) is as follows:
  • The x-axis divides the pattern into two specular halves. We also distinguish left and right regions.

  • The attractors possess a vertically extended region showing more density of points than the rest of regions at its right and left sides. This central structure will be named “Head”.

  • The structures on the left side of the Head located in the central region will be named “Body”.

  • The structures at the region on the right side of the Head will be named “Antennae”.

  • Finally, the structures at the most external regions on the left side of the Head will be named “Limbs”.

It seems that the global structure is divided into two perfect specular halves located in the positive and negative x-axis at a first sight. Actually, although they have a very high degree of symmetry, they are not fully symmetrical: there are local asymmetries along both halves, e.g., especially distinguishable in some circular patterns at the central regions of the Body and the Head.

We decided to verify, again based on a trial and error basis, if slight variations of the mathematical expression of subfamily (4) lead to similar visual properties following the same principles: resemble organic patterns, resemble mechanisms of segmentation and resemble the bioluminescence patterns of living beings.

Remark 4

Our efforts were focused on the symmetries of the complex plane, on the effects of modifications of the sine and cosine functions and on variations of the exponents of the polynomials of the function g(zt).

These are the most relevant results:
  • Modification of the signs of both the \(\mathfrak {I}\) and \(\mathfrak {R}\) values, generating its three simple specular version: \(+\mathfrak {I}\) and \(-\mathfrak {R}\) (specular reflection on the x-axis), \(-\mathfrak {I}\) and \(+\mathfrak {R}\) (specular reflection on the y-axis) and \(-\mathfrak {I}\) and \(-\mathfrak {R}\) (specular reflection on the x- and y-axes). On each case, a slight variation of the densities (number of points accumulated in a specific region of the attractor) and the locations of the subpatterns happens (Fig. 10). But, in general, the aspect of the whole shape barely changes:
    $$\begin{aligned} T(D)&= {} \left\{ \vphantom{\left( \frac{1}{((x_{n} + y_n i)+1)^{2}}\right)}(x,y): (x_{n+1},y_{n+1})\right. \nonumber \\&= {} \left. \left( \mathfrak {I}\left( \frac{1}{((x_{n} + y_n i)+1)^{2}}\right) \cdot \sin {D}, \right.\right.\\&\quad\left.\left.- \mathfrak {R}\left( \frac{1}{((x_{n} + y_n i)+1)^{2}}\right) \cdot \cos {D}\right) \right\} , \end{aligned}$$
    (6)
    $$\begin{aligned} U(D)&= {} \left\{\vphantom{\left( \frac{1}{((x_{n} + y_n i)+1)^{2}}\right)} (x,y): (x_{n+1},y_{n+1}) \right. \nonumber \\&= {} \left. \left( - \mathfrak {I}\left( \frac{1}{((x_{n} + y_n i)+1)^{2}}\right) \cdot \sin {D},\right.\right.\\&\quad\left.\left. \mathfrak {R}\left( \frac{1}{((x_{n} + y_n i)+1)^{2}}\right) \cdot \cos {D}\right) \right\} , \end{aligned}$$
    (7)
    $$\begin{aligned} V(D)&= {} \left\{\vphantom{\left( \frac{1}{((x_{n} + y_n i)+1)^{2}}\right)} (x,y): (x_{n+1},y_{n+1})\right. \nonumber \\&= {} \left. \left( - \mathfrak {I}\left( \frac{1}{((x_{n} + y_n i)+1)^{2}}\right) \cdot \sin {D}, \right.\right.\\&\quad\left.\left.- \mathfrak {R}\left( \frac{1}{((x_{n} + y_n i)+1)^{2}}\right) \cdot \cos {D}\right) \right\} . \end{aligned}$$
    (8)
  • Modification of the signs of the formulation such that the points are relocated to their specular y-reflection only when their real part is positive. The result is a new pattern (Fig. 11):
    $$\begin{aligned} W(D) & = \{ (x,y):(x_{{n + 1}} ,y_{{n + 1}} ) \\ & \; = \left\{ {\begin{array}{*{20}l} {{\text{ when }}\Re (\frac{1}{{((x_{n} + y_{n} i) + 1)^{2} }}) \ge 0} \hfill \\ {{\text{ then }}} \hfill \\ {( - \Im (\frac{1}{{((x_{n} + y_{n} i) + 1)^{2} }})\cdot\sin D, - \Re (\frac{1}{{((x_{n} + y_{n} i) + 1)^{2} }})\cdot\cos D),} \hfill \\ {{\text{ else }}} \hfill \\ {(\Im (\frac{1}{{((x_{n} + y_{n} i) + 1)^{2} }})\cdot\sin D,\Re (\frac{1}{{((x_{n} + y_{n} i) + 1)^{2} }})\cdot\cos D)} \hfill \\ \end{array} } \right. \\ \end{aligned}$$
  • Modification of the signs of the formulation such that the points are relocated at their specular reflection depending on the quadrant where the points are initially located. The result is a new pattern (Fig. 12):
    $$\begin{aligned} X(D) & = \{(x,y): (x_{n+1},y_{n+1}) \nonumber \\ & = {\left\{ \begin{array}{ll} \text { when } \\ \mathfrak {R}(\frac{1}{((x_{n} + y_n i)+1)^{2}}) \ge 0, \mathfrak {I}(\frac{1}{((x_{n} + y_n i)+1)^{2}}) \ge 0 \\ \wedge \\ \mathfrak {R}(\frac{1}{((x_{n} + y_n i)+1)^{2}})< 0, \mathfrak {I}(\frac{1}{((x_{n} + y_n i)+1)^{2}}) < 0\\ \text { then } \\ (\mathfrak {I}(\frac{1}{((x_{n} + y_n i)+1)^{2}})\cdot \sin {D}, - \mathfrak {R}(\frac{1}{((x_{n} + y_n i)+1)^{2}})\cdot \cos {D}), &{} \\ \text { else } \\ (-\mathfrak {I}(\frac{1}{((x_{n} + y_n i)+1)^{2}})\cdot \sin {D}, \mathfrak {R}(\frac{1}{((x_{n} + y_n i)+1)^{2}})\cdot \cos {D}) \end{array}\right. } \end{aligned}$$
    (9)
  • Modification of the constant coefficient of the polynomial \(z+1\) included in the denominator of \(g(z,t)=\frac{1}{(z+1)^{t}}\). E.g., \(g(z,t)=\frac{1}{(z+1.1)^{t}}\). The result is a new pattern (Fig. 13):
    $$\begin{aligned} Y(D)&= {} \left\{ \vphantom{\left( \frac{1}{((x_{n} + y_n i)+1.1)^{2}}\right)}(x,y): (x_{n+1},y_{n+1})\right. \nonumber \\&= {} \left. \left( \mathfrak {I}\left( \frac{1}{((x_{n} + y_n i)+1.1)^{2}}\right) \cdot \sin {D}, \right.\right.\\&\quad\left.\left.- \mathfrak {R}\left( \frac{1}{((x_{n} + y_n i)+1.1)^{2}}\right) \cdot \cos {D}\right) \right\} . \end{aligned}$$
    (10)
The Head, Body, Antennae and Limbs regions are clearly distinguishable in these new patterns.

Resemblance to the External Morphology of Some Invertebrates

The patterns that we are considering potentially similar to the external morphology of some families of living beings are based on the set of attractors of each complete subfamily, including both the stable orbits and the chaotic basins of attraction.

We have been able to identify similarities with some structures of invertebrate life forms, especially insects and zooplankton, for different modifications (see “Visual Characteristics, Symmetry–Asymmetry and Modifications of the Model”) of the subfamily of D-bifurcations S(D) (4). Due to the peculiar distribution and accumulation of the clusters of points, the patterns also resemble visually the mechanisms of bioluminescence.
Fig. 14

Patterns resembling the thorax and abdomen of Bembicini wasp (Wikimedia [16]), and the limbs of a tardigrade (Wikimedia [17]), obtained from the subfamily (4)

Fig. 15

Patterns resembling the morphology of the Clogmia albipunctata (moth fly) (Wikimedia [18]), obtained from the subfamily \(B(D,0.9,\frac{1}{(z^{1/5}+1)^{0.9}})\)

Fig. 16

Patterns resembling the morphology of the Acherontia atropos (Wikimedia [19]), obtained from the subfamily (9). The thorax, abdomen and main proportions of the body, especially the patterns near the head, are interestingly similar

Fig. 17

Patterns resembling the morphology and bioluminescence of the (bottom-left) Ctenophora (Sea Walnut) (Wikimedia [22]) and (bottom-right) Mnemiopsis leidyi (Wikimedia [23]) zooplanktons, obtained from the subfamily (10)

These are some examples:
  • Patterns resembling the thorax and abdomen of Bembicini wasp, and the limbs of a tardigrade (Fig. 14).

  • Patterns resembling the morphology of the Clogmia albipunctata (moth fly) (Fig. 15).

  • Patterns resembling the morphology of the Acherontia atropos. The thorax and abdomen and main proportions of the body, especially the patterns near the head, are interestingly similar (Fig. 16).

  • Patterns resembling the morphology and bioluminescence of the Ctenophora and Mnemiopsis leidyi zooplanktons (Fig. 17).

In addition, the following online supplementary materials will help to explore deeply other modifications of the main family (1) exposed in “Formulation of a New Family of Nonlinear Dynamical Systems”.
  • Animations (Marciel [20, 21]) showing the cinematic evolution (using t as the time control parameter) of some subfamilies of attractors.

  • A Python 3.6.3 script (Marciel [24]) able to plot the subfamily of systems defined by expression:
    $$\begin{aligned} S(D)&= {} \left\{ \vphantom{\left( \frac{1}{((x_{n} + y_n i)+1)^{2}}\right)}(x,y): (x_{n+1},y_{n+1})\right. \nonumber \\ & ={} \left. \left( \mathfrak {I}\left( \frac{1}{((x_{n} + y_n i)+1)^{2}}\right) \cdot \sin {D}, \right.\right.\\&\quad\left.\left.\mathfrak {R}\left( \frac{1}{((x_{n} + y_n i)+1)^{2}}\right) \cdot \cos {D}\right) \right\} (4). \end{aligned}$$
    (11)

Concluding Summary

The principal characteristics of our proposed new family of discrete-time dynamical systems are the following:
  • The main abstract family (1) is a broad non-continuous and non-smooth system that can be easily specialized into different subfamilies of discrete-time maps, each one of them presenting very diverse organic-like patterns, obtained from the visualization of their D-bifurcations, seen as a unique, whole visual structure, for specific discrete-time values of time (t).

  • Most of the D-bifurcations arrive to stable orbits, and some of them to chaotic basins of attraction. The orbits are usually clusters of points and spirals distributed along a bounded complex region, e.g., the subfamily S(D) (4) is located at \(z=a+bi\) where \(a,b \in [-5,5]\).

  • The relationship, represented by a escape time algorithm, between the control parameters of the subfamily of D-bifurcations S(D) (4) seems a genuine new kind of complex fractal structure, initially unrelated to previously existing families of fractals. We hypothesize that other modifications of the main abstract family (1), e.g., other specializations of the subfamilies (2) and (3), will have similar fractal relationships. Besides, we also belive that other classical dynamical systems could have this kind of chaotic fractal relationship between some of their control parameters. As far as we are aware, such studies still have not been done, and some new fractal structures might be waiting to be found following a escape time algorithm strategy.

  • The potential capability of generating organic-like patterns and simulating the accumulation of bioluminescence in clusters could contribute as an alternative path for the mathematical formulation and study of the morphology of some families of invertebrates.

The list of open points is still broad. Further study is required in the following topics:
  • Is there a mathematical equivalence or relationship between our pattern models and the classical reaction–diffusion and cell oscillation models associated with the morphology and the mechanisms of segmentation of invertebrates?

  • Taxonomy of the fractal relationship between the control parameters: what type of fractal is it? Is it related to other families of fractals or is it a genuine new kind of fractal structure? Do classical dynamical systems possess these kind of structures?

  • Extensive study of what other combinations of the control parameters and the function g(zt) provide new organic-like patterns and new similarities to other existing families of invertebrates.10

  • The non-continuity and non-smoothness of these families of dynamical systems make it difficult to advance in some areas, e.g, the study of the Hausdorff dimension of the attractors, the study of the Lyapunov exponent, the study of the phase portraits of each subfamily, and a broader study of the local bifurcations on each subfamily.

Footnotes

  1. 1.

    The terms “discrete-time dynamical system” and “map” will be used indistinctly along this manuscript.

  2. 2.

    In the recent literature, the works of Aubin and Dahan [8], and Holmes [9], have been very useful for us to understand the state of the art of the field.

  3. 3.

    E.g., its application to game theory to simulate the biological strategy-oriented evolution of species, presented by Cases and Anchorena in Grana, Duro, d’Anjou and Wang’s book “Information Processing with Evolutionary Algorithms” [10].

  4. 4.

    There are also interesting similarities in other fields of study, e.g., Christoforou et al. [12] have adapted Evolutionary Game Theory to master-working computing, defining a reliability algorithm for Internet-based master–worker task computations, whose iterative formula resembles some of the aforementioned population models described by May and Oster.

  5. 5.

    Tested programatically by applying randomized initial conditions and visualizing the attractors.

  6. 6.

    The patterns are shown in white color over a black background for a better observation of the resemblance to the external morphology of some families of invertebrates and the mechanisms of bioluminescence of some species. The remaining figure (Fig. 4, study of the bifurcation diagrams) is shown using the same rule for a better visualization of the local bifurcations.

  7. 7.

    As stated in the Definition 2 of the present chapter, we have computationally verified that \({\text {Max}}D=8 \times 10^3\) is a good enough maximum value to observe the behavior of the discrete-time family of dynamical systems.

  8. 8.

    This kind of escape time algorithm is used, for instance, to visualize the classical Gamma function fractal.

  9. 9.

    There are other interesting studies about the fractal nature of the morphology of invertebrates. For instance Castrejón et al. [15] evaluated the geometrical complexity of butterflies’ wings through the calculation of the fractal dimension of their patterns.

  10. 10.

    The patterns presented in this paper are just some initial examples. If an automation method is defined, e.g., using NCC (Normalized Correlation Coefficient) based techniques to match the resulting patterns with possible families of insects, new similarities should be found with a better time–effort cost.

Notes

Acknowledgements

The author would like to thank the editor and the reviewers for their valuable suggestions and comments, which greatly improve the quality of the paper. Thanks are also due to Noemí Martín Santo and Celio H. Barreto for the very useful insights and corrections.

Compliance with Ethical Standards

Conflict of interest

David M. Marciel declares that he has no conflict of interest.

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Copyright information

© Springer Nature Singapore Pte Ltd 2019

Authors and Affiliations

  1. 1.TokyoJapan

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