# Abstract #

We characterize, by periodicities, some dynamic systems represented by maps. This is an alternative method to the common bifurcation diagrams calculated by using the exponents of Lyapunov and allows us to visualize the typical structures in the parameter space, such as the "shrimp", but also with the details of the oscillatory regimes that could be important from a practical point of view point of view.

# 1 . INTRODUCTION #

A very important aspect in the study of dynamic systems is that concerning those that have non-linear character which allows to study not only issues related to stability but also possible chaotic behaviors. Dynamic systems can be represented either by maps (difference equations) or by continuous flows (differential equations). When systems are non-linear, their study is often handled numerically since in general it is difficult to find analytical solutions. Aspects such as bifurcations, periodicities and chaos are a basic part of the so-called Nonlinear Dynamics and there are several ways to address them.

<center>http://confluencia.network/wp-content/uploads/2016/11/Sin-t%C3%ADtulo4.png</center>
<center>[Source](http://confluencia.network/wp-content/uploads/2016/11/Sin-t%C3%ADtulo4.png)</center>


The article is organized as follows: Section 2 introduces the logistic map and describes the periodicities in it. In Section 3 the Hénon map and mainly the structure of its parameter space are analyzed by means of periodicities. In Section 4 some regions of the Tinkerbel map parameter space are analyzed, where structures of periodicity different from the typical "shrimp" are found. In Section 5 we study the behavior in the parameter space of a neuron model, in which the role played by periodicities seems to be important. Finally, Section 6 gives the conclusions and perspectives of the research carried out.



# 2 . THE LOGISTICS MAP #

The so-called logistic equation emerged a long time ago as an alternative model of population growth different from the Malthusian one that has an exponential character. Its first formulation is due to the Brussels-based mathematician J. F. Verhulst . This model, described by a differential equation, is widely used in demography and ecology . The importance of Verhulst's work is reflected in the tributes he received to commemorate the 200th anniversary of his birth, especially in his hometown where the Verhulst 200 on Chaos conference took place, from whose memoirs a book was published with a detailed review of the logistic equation and its applications . However, this model can be reduced to a map and in this way, be analyzed in a simpler but at the same time deeper way. The logistic map can be expressed in the form:

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image001.png</center>

where x is the dynamic variable and μ is the control parameter. The logistic map analysis is one of the basic elements of nonlinear dynamics because it allows to study behaviors that range from stationarity, passing through increasingly complicated periodic behaviors until reaching chaos (absence of periodicity), only varying the control parameter .

Exactly, the term chaos is introduced in 1975 by reason of a study of the logistic map . Numerous studies were carried out using this simple model that nevertheless presents a complicated dynamic . Particularly, the chaotic aspect that this system presents for certain μ values ​​has been approached under different perspectives; from simple calculations to show the emergence of period 3 cycles as a consequence of a tangent bifurcation that emerges after a chaotic behavior; until the development of new concepts such as intermittency . On the other hand, starting from the analysis of the cascades of period doubling in the logistic map, some results could be generalized and universalized. In Fig. 1 three forms of characterization of the logistic map are shown: (a) bifurcation diagram, (b) exponents of Lyapunov and (c) periodicities. All these representations clearly show us the difference between periodic and chaotic behavior; thus, in Fig. 1 (a) the cascades of period splitting are identified and the dark regions of the diagram represent situations of chaos. In Fig. 1 (b), the periodic regions differ from the chaotic ones in that for the periodic ones, the exponent of Lyapunov is negative (λ ≤ 0), whereas for the chaotic ones, it is positive (λ> 0). The distribution of periodicities is shown as a kind of steps in Fig. 1 (c), with step zero corresponding to the chaotic behavior of the system.

The interesting thing about working with periodicities lies in the fact of identifying more finely the oscillatory behavior of the system, which is not very clear in the other representations. However, a disadvantage is that in order to have a good determination of the periodicities, a long iteration process is necessary; The foregoing also applies to the calculation of Lyapunov exponents since the stability of these must be achieved.


# 3  . THE MAP OF HENON #

In 1976, Hénon proposes a reductionist model able to reproduce in a computationally simpler way the results from the paradigmatic model of Lorenz . In its simplest form it can be expressed as:

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image002.png</center>

This transformation has as one of its most important characteristics that of preserving the area in the phase space when | b | = 1 since J = 1; being J, the Jacobian determinant, which means that under this condition, the system he describes is conservative. A large number of studies have been carried out on the Hénon map, among which we can highlight the analysis of period splits , the analysis of fixed points that give rise to limit cycles with different periods ; the exhaustive exploration in the parameter space where the "shrimp" structures are located ; the analysis with isoperiodic diagrams  and the basins of attraction , among others.

The route to chaos in the Hénon map can occur through a cascade of period unfolding as shown in Fig. 2 where the value of parameter a is set and b is varied. The same characterization is done as for the logistic map and some important aspects are observed such as bifurcations with edge collisions similar to those exposed in al. We will return to this aspect in the study of the Tinkerbell map.

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image003.gif</center>
<center>[Source](http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image003.gif)</center>
##### <center>Figure 1: (Color online) Dynamic characterization of the logistic map as a function of the control parameter μ, by (a) a bifurcation diagram, (b) the exponents of Lyapunov and (c) the periodicities.</center> #####


<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image004.gif</center>
<center>[Source](http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image004.gif)</center>
##### <center>Figure 2: (Color online) Dynamic characterization of the Hénon map as a function of the control parameter b, by (a) a bifurcation diagram, (b) the exponents of Lyapunov and (c) the periodicities, when a = 1.5.</center> #####

In Fig. 3, we show a phase diagram for the Hénon map, considering the largest exponents of Lyapunov (a) and the periodicities (b). Using the same region that is reported in. As you can see, the structures that denote periodic behavior have the form of "shrimp". From the diagram obtained from the largest exponents of Lyapunov, although some regions of superestability can be identified inside the shrimp, apart from that, the periodicities that correspond to each of these structures can not be identified, which is easily achieved if the periodicity diagram is used. It is interesting to note that the shrimp are not completely isoperiodic and that at the edges of them there are splittings of period that naturally lead to the chaotic region.

# 4  . MAPA TINKERBELL #

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image005.gif</center>
<center>[Source](http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image005.gif)</center>
### <center>(a)</center> ###

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image006.gif</center>
<center>[Source](http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image006.gif)</center>
### <center>(b)</center> ###

##### <center>Figure 3: (Online color) Parameter space of the Hénon map obtained from (a) the largest exponents of Lyapunov and (b) the periodicities; in this case, each color of the bar on the right corresponds to a certain periodicity; the situations of chaos and stationarity are indicated by the colors of the lower boxes (blue and white respectively) and the situation in which the periodicity is greater than or equal to 25 by the black color, corresponding to the upper frame of the bar.</center> #####

The Tinkerbell map appears in general as an academic example of a dynamic system. However, given its characteristics, it offers a dynamic richness that can be exploited. The explicit form of this map is:

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image007.png</center>


One of its characteristics is to present a fractal structure at the borders of the basin of attraction, as seen in Fig. 4. The shape of the basin of attraction is geometrically the same for other values ​​of the parameters despite that the behavior may turn out to be totally different. Thus, one can have a completely chaotic basin like that of Fig. 4 (a) or a completely periodic basin with other parameter values, as shown in Fig. 4 (b).

On the other hand, since there are 4 control parameters in the model, different sections of the parameter space can be obtained, each of which has interesting aspects to study. For simplicity, we will focus on the section of the parameter space (a, b) shown in Fig. 5. As can be seen in Fig. 4, there are many regions in which there is divergence; however, you can locate regions in which there are periodic structures surrounded by chaos. In Fig. 5 (a) two of these regions are separated by a region where the behavior of the system is stationary.

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image008.jpg</center>
<center>[Source](http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image008.jpg)</center>

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image009.jpg</center>
<center>[Source](http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image009.jpg)</center>

##### <center>Figure 4: (Online color) Basins of attraction (a) chaotic for the Tinkerbell model when the values of the parameters are: a = 0.5, b = -0.6, c = 2.2 and d = 0.5. (b) periodic with a = -0.51, b = -0.9, c = 2.2 and d = 0.5. The red regions correspond to periodicities of order 5 while the yellow regions are of order 15. In both cases, the pink zone indicates that for these parameter values, there is no convergence.</center> #####

Also, it seems that there seems to be a kind of connection between structures of equal periodicity, which is highlighted by arrows for the case of periodicities 5 and 7. It is interesting to observe in Fig. 5 (b), one of these regions, in which shows the presence of shrimp sequences and other periodic structures that indicate degenerated routes to chaos similar to those found in. In particular, the sequence that goes from periodicity 1 to ∞ and that is shown with the white arrow. Of course, there are other sequences considering these same multiperiodic structures. Thus, we will have, for example, sequences:

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image007.png</center>

The way in which the periodic regions are distributed leads us to think that the route to chaos does not necessarily occur through a cascade of period doubling and that other bifurcations like the border collisions mentioned above take place. when analyzing the map of Hénon. You can also see the existence of minor periodic structures whose periodicities also seem to follow a certain sequence. Finally, it should also be noted that the periodic structures found in the Tinkerbell map have a morphology different from that of the "shrimp" found in most dynamic systems, both discrete and continuous.

# 5 .  A MODEL OF NEURONA #

As is well known, the brain - in particular humans - is one of the most difficult complex systems to analyze and although in recent years considerable progress has been made in Neuroscience, there are still aspects that remain obscure in relation to its functionality. The brain has as fundamental parts the neurons that can reach in number to 10 ^ 11. These neurons are connected to each other forming highly complex networks because the number of links that each neuron can have can reach 10 ^ 4.

There are different types of neurons and for many years we have tried to model them both individually and when they connect to others. A neuron is considered as a system that is not in balance and also has several mechanisms of feedback and delay, which allow the oscillatory nature of it. One of the characteristics of the behavior of neurons is to have electrical excitability. On the other hand, the resting potential and the action potential are highlighted as typical in neuronal functioning. Since there are many types of neurons and each with different characteristics, different models have been proposed for the description of neurons. The first model proposed was that of Hodgkin-Huxley in 1952, which was the culminating part of a series of experimental works with giant squid neurons.

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image011.jpg</center>
<center>[Source](http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image011.jpg)</center>

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image012.jpg</center>
<center>[Source](http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image012.jpg)</center>

##### <center>Figure 5: (Color online) Parameter space (a, b) for the Tinkerbell map using periodicities for its characterization. (a) Two regions that present periodic structures and chaos, separated by a region of stationarity. (b) Magnification of the region framed in (a), where well-defined sequences are observed in the larger structures and others for the smaller ones. The white arrow indicates a sequence in which the larger periodic regions participate .. A color code similar to that in Fig. 3 is used to denote the periodicities, where we emphasize that the blue region corresponds to chaos, the white one to stationarity and the rose to divergence.</center> #####

Later, other models that tried to express in a simpler way the equations that govern the behavior of neurons were postulated. All these models consist of systems of differential nonlinear equations, so their analytical solution is almost impossible. Thus, in general, the problem is addressed numerically. With the aim of simplifying these models, Rulkov proposes a model consisting of a map with two variables that allows rescuing the most important behaviors in neurons, such as the fact of having sustained oscillations, situations of stationarity and the so-called bursts or "bursts" consisting of oscillatory episodes of type "spikes" (peaks) followed by phases of stationarity or "silence". The model is expressed by:

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image013.png</center>

where x and y are the fast and slow dynamic variables respectively when the μ parameter takes small values such as μ = 0.001. Following the typical values considered in, 3 regions can be determined in the parameter space (σ, α), as shown in Fig. 6; These parameters are what determine the behavior of the neuron and are related to the external influences applied. In the boxes, the typical behaviors of the variable x are observed.

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image014.jpg</center>
<center>[Source](http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image014.jpg)</center>

##### <center>Figure 6: (Color online) Regions in the parameter space (σ, α), where the regions of stationarity or silence are distinguished, the one of sustained oscillations (spikes) and the one of burst of spikes, where the Boxes show examples of the time evolution of variable x in each of these regions.</center> #####

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image015.jpg</center>
<center>[Source](http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image015.jpg)</center>

##### <center>Figure 7: (Color online) Periodicities in the parameter space of the Rulkov model. A color code similar to that of Fig. 3 is used in order to identify the regions of stationariness and chaos.</center> #####

As can be seen, the determination of the boundary between the region of "burst of spikes" and the region of continuous peaks (sustained oscillations) is not very well defined; in particular, the lower region of the "peak burst" could more properly be considered as a region of oscillations in which there is an impulse as if it were a Dirac delta function and which, as will be explained later in the Fig. 9 (a), it is a region characterized by a chaotic behavior. For a finer analysis, we proceed in Fig. 7 to study in greater detail this area of ​​the parameter space.

We also draw attention to the fact that the borders reported in do not fully coincide with those we determine by exhaustively scanning the values ​​of the parameters. As can be seen in Fig. 7, the periodicities in the region of sustained oscillations are well defined and decrease in steps of 1 after the system leaves the chaotic window. It is interesting to note also that the areas corresponding to smaller periodicities increase as the periodicity decreases, which is coherent if one thinks that the route to chaos implies increasingly narrow windows of periodicity. To have a clearer idea of ​​how the bifurcation occurs, it is represented in Fig. 8, the same dynamic characterization used in Figs. 1 and 2, where it is interesting to observe the change from stationarity to chaos without mediating a cascade of period unfolding.

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image016.jpg</center>
<center>[Source](http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image016.jpg)</center>

##### <center>Figure 8: (Color online) Dynamic characterization for the Rulkov model as a function of the control parameter σ, by (a) a bifurcation diagram, (b) the exponents of Lyapunov and (c) the periodicities, .when α = 2.5 and μ = 0.001.</center> #####

To finish our analysis of the neuron model, we concentrate on the region where there are bursts of peaks because in some sectors of the same chaotic behavior occurs. In Fig. 9 (a) this region is represented by exponents of Lyapunov and in Fig. 9 (b) through an analysis counting the number of peaks of the bursts, when these are periodic. It is observed that the number of peaks in the bursts tends to increase when the parameters σ and α grow. We must also mention that by varying the value of the parameter μ for the same regions of the parameter space (σ, α), it is observed that the chaotic region is greater when the value of μ decreases.

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image017.jpg</center>
<center>[Source](http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image017.jpg)</center> 

<center>http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image018.jpg</center>
<center>[Source](http://www.scielo.org.bo/img/revistas/rbf/v19n19/v19n19a01-image018.jpg)</center>

##### <center>Figure 9: (Color online) Parameter space for the Rulkov model considering (a) the exponents of Lyapunov. The color bar indicates the value of the largest exponents of Lyapunov. (b) The number of peaks per burst; In black, the chaotic regions are represented and in colors the regions where there is periodicity. The white region means there are no bursts of peaks.</center> #####

# CONCLUSIONS AND PERSPECTIVES #

In the first place, the pertinence of the periodicity calculation was verified as a useful and simple alternative for the characterization of dynamic systems.

Comparing the characterizations by exponents of Lyapunov and periodicities, one of the advantages of working with the latter is to be able to discriminate each order of periodicity and verify the possible sequences of them that allow to have a clearer idea of ​​how the path to the chaos. However, one of the problems of working with periodicities is that of having to consider quite long times in the regions where bifurcations occur, otherwise, erroneous results are obtained that seem to show chaos in all borders between periodicities.

The analysis of Rulkov's neuron model shows interesting results because it allows us to identify 3 zones with well-defined characteristics. In the region of sustained oscillations, the route to chaos occurs through a cascade but not with a period splitting but with a discrete increase in periodicity (in steps of one). On the other hand, the exit from chaos is directly towards stationarity. Additionally, the analysis in the region where there are bursts of peaks, indicates that there are chaotic regions that we can identify by calculating exponents of Lyapunov and also, you can also describe the behavior of these bursts by quantifying the number of peaks by Burst when these bursts are periodic. This is important since it allows choosing the behavior according to the system studied. This type of behavior is also observed in some species of male fireflies and this model could eventually be applied to study the emission of flashes of these insects. Another application of this kind of model can be found in systems that describe biological rhythms in which bursts of peaks occur, such as those mentioned in and more specifically in relation to Ca2 + oscillations. It is important to study similar maps not only individually but also forming networks just as neurons do.

References

[11996Abarbanel et al.Abarbanel, Rabinovich, Selverston, Bazhenov, Huerta, Sushchik, & Rubchinsky](http://www.scielo.org.bo/rbf19html/articulos/ramirez/ramirez.html#CITE1)

1.- Abarbanel, H.D., Rabinovich, M.I., Selverston, A., Bazhenov, M.V., Huerta, R., Sushchik, M.M., & Rubchinsky, L.L. 1996, Physics-Uspekhi 39, 337 [Links]
[21989Albahadily et al.Albahadily, Ringland, & Schell](http://www.scielo.org.bo/rbf19html/articulos/ramirez/ramirez.html#CITE2)

2. Albahadily, F. N., Ringland, J., & Schell, M. 1989, The Journal of Chemical Physics 90, 813 [Links]
[31996 Alligood et al.Alligood, Sauer, & Yorke](http://www.scielo.org.bo/rbf19html/articulos/ramirez/ramirez.html#CITE3)

3. Alligood, K., Sauer, T. D., & Yorke, J. A. 1996, Chaos: An Introduction to Dynamical Systems (Springer-Verlag, New York) [Links]
[41994 Argyris et al. Argyris, Faust, & Haase](http://www.scielo.org.bo/rbf19html/articulos/ramirez/ramirez.html#CITE4)

4.- Argyris, J., Faust, G., & Haase, M. 1994, An Exploration to Chaos (Elsevier Science B.V., Amsterdam) [Links]
[52005Ausloos & Dirickx](http://www.scielo.org.bo/rbf19html/articulos/ramirez/ramirez.html#CITE5)

4.- Ausloos, M. & Dirickx, M. 2005, The Logistic Map: Map and the Route to Chaos: From the Beginning to Modern Applications (Springer-Verlag, Heidelberg) [Links]
[61996Bechhoefer](http://www.scielo.org.bo/rbf19html/articulos/ramirez/ramirez.html#CITE6)

5.- Bechhoefer, J. 1996, Mathematics Magazine 69, 115 [Links]
[71981Bountis](http://www.scielo.org.bo/rbf19html/articulos/ramirez/ramirez.html#CITE7)

6.- Bountis, T. C. 1981, Physica D 3, 577 [Links]
[81993Cabral et al.Cabral, Lago, & Gallas](http://www.scielo.org.bo/rbf19html/articulos/ramirez/ramirez.html#CITE8)

[382011Ramírez et al.Ramírez, Deneubourg, Guisset, Wessel, & Kurths](http://www.scielo.org.bo/rbf19html/articulos/ramirez/ramirez.html#CITE46)

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