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Nonlin. Processes Geophys., 25, 291–300, 2018

https://doi.org/10.5194/npg-25-291-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 4.0 License.

Nonlinear analysis of the occurrence of hurricanes in the Gulf of

Mexico and the Caribbean Sea

Berenice Rojo-Garibaldi1, David Alberto Salas-de-León2, María Adela Monreal-Gómez2, Norma

Leticia Sánchez-Santillán3, and David Salas-Monreal4

1Posgrado en Ciencias del Mar y Limnología, Universidad Nacional Autónoma de México, Av. Universidad 3000,

Col. Copilco, Del. Coyoacan, Cd. Mx. 04510, Mexico

2Instituto de Ciencias del Mar y Limnología, Universidad Nacional Autónoma de Mexico, Av. Universidad 3000,

Col. Copilco, Del. Coyoacan, Cd. Mx. 04510, Mexico

3Departamento El Hombre y su Ambiente, Universidad Autónoma Metropolitana, Calz. del Hueso 1100, Del. Coyoacán,

Villa Quietud, Cd. Mx. 04960, Mexico

4Instituto de Ciencias Marinas y Pesquerías, Universidad Veracruzana, Hidalgo No. 617, Col. Río Jamapa,

C.P. 94290 Boca del Rio, Veracruz, Mexico

Correspondence: David Alberto Salas-de-León (dsalas@unam.mx)

Received: 21 September 2017 – Discussion started: 6 October 2017

Revised: 4 April 2018 – Accepted: 5 April 2018 – Published: 27 April 2018

Abstract. Hurricanes are complex systems that carry large

amounts of energy. Their impact often produces natural dis-

asters involving the loss of human lives and materials, such as

infrastructure, valued at billions of US dollars. However, not

everything about hurricanes is negative, as hurricanes are the

main source of rainwater for the regions where they develop.

This study shows a nonlinear analysis of the time series of

the occurrence of hurricanes in the Gulf of Mexico and the

Caribbean Sea obtained from 1749 to 2012. The construc-

tion of the hurricane time series was carried out based on

the hurricane database of the North Atlantic basin hurricane

database (HURDAT) and the published historical informa-

tion. The hurricane time series provides a unique historical

record on information about ocean–atmosphere interactions.

The Lyapunov exponent indicated that the system presented

chaotic dynamics, and the spectral analysis and nonlinear

analyses of the time series of the hurricanes showed chaotic

edge behavior. One possible explanation for this chaotic edge

is the individual chaotic behavior of hurricanes, either by cat-

egory or individually regardless of their category and their

behavior on a regular basis.

1 Introduction

Hurricanes have been studied since ancient times, and their

activity is related to disasters and loss of life. In recent years,

there has been considerable progress in predicting their tra-

jectory and intensity once tracking has begun, as well as

their number and intensity from one year to the next. How-

ever, their long-term and very short-term prediction remains

a challenge (Halsey and Jensen, 2004), and the damage to

both materials and lives remains considerable. Therefore, it

is important to make a greater effort regarding the study of

hurricanes in order to reduce the damage they cause. The

periodic behavior of hurricanes and their relationships with

other natural phenomena have usually been performed with

linear-type analyzes, which have provided valuable informa-

tion. However, we decided to make a different contribution

by carrying out a nonlinear analysis of a time series of hurri-

canes that occurred in the Gulf of Mexico and the Caribbean

Sea, as the dynamics of the system are controlled by a set

of variables of low dimensionality (Gratrix and Elgin, 2004;

Broomhead and King, 1986).

One of the core sections of this work was the elaborate

time series that was built, especially for the oldest part of the

registry, for which it was possible to compile a substantial

and robust collection. This provided our time series with an

amount of data with which it was possible to perform the

Published by Copernicus Publications on behalf of the European Geosciences Union & the American Geophysical Union.

292 B. Rojo-Garibaldi et al.: Nonlinear analysis of the occurrence of hurricanes

desired analysis; otherwise, it would have been impossible to

study this natural phenomenon via nonlinear analysis.

Different methods have been used in the analysis of non-

linear, non-stationary and non-Gaussian processes, includ-

ing artiﬁcial neural networks (ASCE Task Committee, 2000;

Maier and Dandy, 2000; Maier et al., 2010; Taormina et al.,

2015). Chen et al. (2015) use a hybrid neural network model

to forecast the ﬂow of the Altamaha River in Georgia; Gho-

lami et al. (2015) simulate groundwater levels using den-

drochronology and an artiﬁcial neural network model for the

southern Caspian coast in Iran. Furthermore, theories of de-

terministic chaos and fractal structure have already been ap-

plied to atmospheric boundary data (Tsonis and Elsner, 1988;

Zeng et al., 1992), e.g., to the pulse of severe rain time se-

ries (Shariﬁ et al., 1990; Zeng et al., 1992) and to tropical

cyclone trajectory (Fraedrich and Leslie, 1989; Fraedrich et

al., 1990). Natural phenomena occur within different con-

texts; however, they often exhibit common characteristics,

or may be understood using similar concepts. Deterministic

chaos and fractal structure in dissipative dynamical systems

are among the most important nonlinear paradigms (Zeng

et al., 1992). For a detailed analysis of deterministic chaos,

the Lyapunov exponent is utilized as a key point and several

methods have been developed to calculate it. It is possible to

deﬁne different Lyapunov exponents for a dynamic system.

The maximal Lyapunov exponent can be determined without

the explicit construction of a time-series model. A reliable

characterization requires that the independence of the em-

bedded parameters and the exponential law for the growth of

distances can be explicitly tested (Rigney et al., 1993; Rosen-

stein et al., 1993). This exponent provides a qualitative char-

acterization of the dynamic behavior and the predictability

measurement (Atari et al., 2003). The algorithms usually em-

ployed to obtain the Lyapunov exponent are those proposed

by Wolf (1986), Eckmann and Ruelle (1992), Kantz (1994)

and Rosenstein et al. (1993). The methods of Wolf (1986)

and Eckmann and Ruelle (1992) assume that the data source

is a deterministic dynamic system and that irregular ﬂuctu-

ations in time-series data are due to deterministic chaos. A

blind application of this algorithm to an arbitrary set of data

will always produce numbers, i.e., these methods do not pro-

vide a strong test of whether the calculated numbers can ac-

tually be interpreted as Lyapunov exponents of a determinis-

tic system (Kantz et al., 2013). The Rosenstein et al. (1993)

method follows directly from the deﬁnition of the Lyapunov

maximal exponent and is accurate because it takes advantage

of all available data. The algorithm is fast, easy to imple-

ment and robust to changes in the following quantities: em-

bedded dimensions, data set size, delay reconstruction and

noise level. The Kantz (1994) algorithm is similar to that of

Rosenstein et al. (1993).

We constructed a database of occurrences of hurricanes

in the Gulf of Mexico and the Caribbean Sea to perform a

nonlinear analysis of the time series, the results from which

can aid in the construction of hurricane occurrence models,

Figure 1. Hurricanes between 1749 and 2012. The dashed line

shows the linear trend (after Rojo-Garibaldi et al., 2016).

which in turn will help to reinforce prevention measures for

this type of hydrometeorological phenomenon.

2 Materials and methods

2.1 Data set description

A detailed analysis of historical reports was carried out in

order to obtain the annual time series of hurricane occur-

rence, from category one to ﬁve on the Safﬁr–Simpson scale,

in the study region from 1749 to 2012. The time series was

composed using the historical ship track of all vessels sail-

ing close to registered hurricanes, the aerial reconnaissance

data for hurricanes since 1944 and the hurricanes reported by

Fernández-Partagas and Díaz (1995a, b, 1996a, b, c, 1997,

1999). All of the abovementioned information in addition to

the database of the HURDAT re-analysis project (HURDAT

is the ofﬁcial record of the United States for tropical storms

and hurricanes occurring in the Atlantic Ocean, the Gulf of

Mexico and the Caribbean Sea) was used in a comparative

way in order to build our time series (Fig. 1), which is cur-

rently the longest time series of hurricanes for the Gulf of

Mexico and the Caribbean Sea. This makes our series ideal

for performing a nonlinear analysis, which would be impos-

sible with the records available in other regions.

Historical hurricanes were included only if they were re-

ported in two or more databases and met both of the fol-

lowing criteria: the reported hurricanes that touched land and

those that remained in the ocean; on the other hand, the fol-

lowed hurricanes were studied considering their average du-

ration and their maximum time (9 and 19 days, respectively).

This was done in order to avoid counting more than one spe-

ciﬁc hurricane reported in different places within a short pe-

riod time; to do this, we followed the proposed method by

Rojo-Garibaldi et al. (2016).

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B. Rojo-Garibaldi et al.: Nonlinear analysis of the occurrence of hurricanes 293

Figure 2. Phase diagrams corresponding to the time series of hurricanes that occurred between 1749 and 2012 in the Gulf of Mexico and the

Caribbean Sea. The xaxis in the four plots indicates the time lag (τ).

2.2 Data reduction and procedures

Before performing the nonlinear analysis of the time series,

we removed the trend; thus, the series was prepared accord-

ing to what is required for this type of analysis. To uncover

the properties of the system, however, requires more than

just estimating the dimensions of the attractor (Jensen et al.,

1985); therefore, three methods were applied in this study:

1. The Hurst exponent is a measure of the independence

of the time series as an element to distinguish a fractal

series. It is basically a statistical method that provides

the number of occurrences of rare events and is usually

called re-scaling (RS) rank analysis (Gutiérrez, 2008).

According to Miramontes and Rohani (1998), the Hurst

exponent also provides another approximation that can

be used to characterize the color of noise, and could

therefore be applied to any time series. The RS helps to

ﬁnd the Hurst exponent, which provides the numerical

value which makes it possible to determine the autocor-

relation in a data series.

2. The Lyapunov exponent is invariant under soft transfor-

mations, because it describes long-term behavior, pro-

viding an objective characterization of the correspond-

ing dynamics (Kantz and Schreiber, 2004). The pres-

ence of chaos in dynamic systems can be solved using

this exponent, as it quantiﬁes the exponential conver-

gence or divergence of initially close trajectories in the

state space and estimates the amount of chaos in a sys-

tem (Rosenstein et al., 1993; Haken, 1981; Wolf, 1986).

The Lyapunov exponent (λ) can take one of the fol-

lowing four values: λ< 0, which corresponds to a sta-

ble ﬁxed point; λ=0, which is for a stable limit cycle;

0 < λ<∞, which indicates chaos; and λ= ∞, a Brown-

ian process, which agrees with the fact that the entropy

of a stochastic process is inﬁnite (Kantz and Schreiber,

2004).

3. The iterated function analysis (IFS) is an easier and sim-

pler way to visualize the ﬁne structure of the time series

because it can reveal correlations in the data and help to

characterize its color, referring color to the type of noise

(Miramontes et al., 2001). Together with the Lyapunov

exponent, the phase diagrams, the false close neighbors

method, the space-time separation plot, the correlation

integral plot and the correlation dimension were taken

into account, the latter two to identify whether the sys-

tem attractor was a fractal type or not. It is important

to compute the Lyapunov exponent, so we used the al-

gorithms proposed by Kantz (1994) and Rosenstein et

al. (1993) to do so.

3 Results and discussions

Figure 1 shows the evolution of the number of hurricanes

from 1749 to 2012 and the linear trend. To have a qualita-

tive idea of the behavior of the number of hurricanes that

occurred in the Gulf of Mexico and the Caribbean Sea from

1749 to 2012, a phase diagram was created using the ”de-

lay method” (Fig. 2). This was also used to elucidate the

time lag for an optimal embedding in the data set. The op-

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294 B. Rojo-Garibaldi et al.: Nonlinear analysis of the occurrence of hurricanes

Figure 3. The mutual information method (a): the xaxis indicates the time lag against the mutual information index (AMI) and the arrow

indicates the ﬁrst, most pronounced minimum with a value of τ=9. The autocorrelation function (b), the xaxis indicates the time lag versus

the value of the autocorrelation function, and the arrow denotes where the ﬁrst zero of the function τ=10 was obtained.

timal time lag (τ) obtained visually from Fig. 2 was equal to

nine, since it was the time in which the curves of the system

were better divided. We must not forget that this was only

a visual inspection, and the delay time is obtained quantita-

tively by other methods. In our case, the hurricane dynamics

were not distinguished through the phase diagram; however,

as any hurricane trajectory starts at a close point location on

the attractor data set which diverges exponentially, the phase

diagram is a primary evidence of a chaotic motion according

to Thompson and Stewart (1986).

The most robust method to identify chaos within the sys-

tem is the Lyapunov exponent. Prior to obtaining the expo-

nent, it was necessary to calculate the time lag and the em-

bedding dimension, and for the latter, the Theiler window

was used. The time lag was obtained via three different meth-

ods:

1. The method of constructing delays, which is observed

visually in Fig. 2.

2. The method of mutual information, which yields a more

reliable result as it takes nonlinear dynamic correlations

into account; in this study, the delay time was obtained

by taking the ﬁrst minimum of the function – in this case

τ=9.

3. The autocorrelation function method, which is based

solely on linear statistics (Fig. 3).

There are two ways to obtain the time lag from the autocor-

relation function:

1. the ﬁrst zero of the function, and

2. the moment in which the autocorrelation function de-

cays as 1/e (Kantz and Schreiber, 2004).

We used the criterion of the ﬁrst zero because the Hurst expo-

nent (H=0.032) indicated that it was a short memory pro-

cess; therefore, the criterion of the ﬁrst zero is the optimal

Figure 4. False close neighbors with a time lag of 10, where the em-

bedding dimension of 5 has a 9.4 % and the embedding dimension

of 4 has a 16.66 % false close neighbors (lower line). False close

neighbors with a time lag of 9, where the embedding dimension of

5 has a 20.15 % and the embedding dimension of 4 has a 20.12 %

false close neighbors (upper line). The values in each line indicate

the optimal dimension for each lag.

method in this type of case. Using this method, the value that

was obtained was τ=10. The value of this parameter is very

important, because if it turns out to be very small, then each

coordinate is almost the same and the reconstructed trajec-

tories look like a line (the phenomenon is known as redun-

dancy). If the delay time is quite large, however, then due

to the sensitivity of the chaotic movement, the coordinates

appear to be independent and the reconstructed phase space

looks random or complex (a phenomenon known as irrele-

vance) (Bradley and Kantz, 2015).

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B. Rojo-Garibaldi et al.: Nonlinear analysis of the occurrence of hurricanes 295

The Hurst exponent helps us to identify the criteria to ﬁnd

a time lag, and also describes the system behavior (Quintero

and Delgado, 2011). This could indicate that the system does

not have chaotic behavior; however, the remaining methods

have indicated the opposite, and as previously mentioned,

the Lyapunov exponent is considered the most appropriate

method for this type of data set. Therefore, different methods

will provide different results, but the time series will indicate

the best method and the result we should use.

It was possible to observe the difference in the time lag

obtained through the autocorrelation function and the mu-

tual information; however, it is necessary to use only one re-

sult. Through the space-time separation graphic and the false

close neighbors method, we obtained embedding dimensions

of m=4 for a τ=9 and m=5 for τ=10, and the Theiler

window with a value of W=16 for τ=9 and W=18 for

τ=10 (Fig. 4). The choice of this window is very important

so as not to obtain subsequent spurious dimensions in the at-

tractor. According to Bradley and Kantz (2015), the Theiler

window ensures that the time spacing between the potential

pairs of points is large enough to represent a distributed sam-

ple identically and independently.

The idea of the false close neighbors algorithm is that at

each point in the time series, Stand its neighbor Sjshould

be searched in a m-dimensional space. Thus, the distance

St−Sj

is calculated iterating both points, given by the

following:

Ri=Si+1−Sj+1

Sl−SJ

.(1)

If Riis greater than the threshold given by Rt, then SJhas

false close neighbors. According to Kennel et al. (1992), a

value of Rt=10 has proven to be a good choice for most data

sets, but a formal mathematical proof for this conclusion is

not known; therefore, if this value does not give convincing

results, it is advisable to repeat the calculations for several

Rt(Perc, 2006). In our case, this value gave relevant results.

It may have some false close neighbors even when work-

ing with the correct embedding dimension. The result of this

analysis may depend on the time lag (Kantz and Schreiber,

2004). In a similar fashion to the delay time, the value of

the embedment dimension is crucial not only for the recon-

struction of the phase space but also to obtain the Lyapunov

exponent. Choosing a large value of mfor chaotic data will

add redundancy and consequently affect the development of

many algorithms such as the Lyapunov exponent (Kantz and

Schreiber, 2004).

The Lyapunov (λ) exponents were obtained using the

Kantz and Rosenstein methods and took the time lag, the

embedding dimension and the Theiler window as the main

values; however, an election of the neighborhood radius for

the exploration of trajectories was also made, as well as the

points of reference and the neighbors near these points. The

modiﬁcation of these parameters is important to corroborate

the invariant characteristic of the Lyapunov exponent. The

Kantz (1994) method using a value of m=4 and τ=9 gave

us an exponent of λ=0.483, while for m=5 and τ=10 the

exponent was λ=0.483. As λis a positive value, it was in-

ferred that our system is chaotic. In addition, the value of λ

obtained for both imbibing dimensions was the same, sug-

gesting that our result is accurate. Using the Rosenstein et

al. (1993) method, the value obtained for m=4 and τ=9

was λ=0.1056, and for m=5 and τ=10, the exponent was

λ=0.112 (Fig. 5).

There was a difference between placing the attractor in an

embedding dimension of m=4 and one of m=5; a better

unfolding of the attractor in the embedding dimension was

observed in m=4 and τ=9. This value of τwas obtained

with the mutual information method, which, according to

Fraser and Swinney (1986) and Krakovská et al. (2015), pro-

vides a better criterion for the choice of delay time than the

value obtained by the autocorrelation function.

It was possible to obtain the correlation dimension D2

(Fig. 6) and the correlation integral (Fig. 6) using the em-

bedding dimension, the delay time and the Theiler window,

following the method of Grassberger and Procaccia (1983a,

1983b). This was done in order to obtain the possible dimen-

sions of the attractor. It should be noted that there is a whole

family of fractal dimensions, which are usually known as

Renyi dimensions, but these are based on the direct appli-

cation of box-counting methods, which demands signiﬁcant

memory and processing and the results of which can be very

sensitive to the length of the data (Bradley and Kantz, 2015).

That is why we chose to use the dimension and integral cor-

relation, which according to Bradley and Kantz (2015) is a

more efﬁcient and robust estimator.

The right panel on Fig. 7 shows the slope trend of the ma-

jority of the slopes of the correlation integral (ε). In the range

of 1 < ε< 10, we are required to have straight lines as an in-

dicator of the self-similar geometry. The value obtained here

corresponds to D2=2.20 which is the aforementioned slope

value. Another method to see the attractor dimension is the

Kaplan–Yorke dimension (DKY), which is associated with

the spectrum of Lyapunov exponents and is given by the fol-

lowing:

DKY =k+Xk

i=1

λi

|λk+1|,(2)

where kis the maximal integer, such that the sum of the k

major exponents is not negative. The fractal dimension with

this method yielded a value of DKY =2.26, which is similar

to the one obtained previously.

Even when all the requirements necessary to apply the

nonlinear analysis to our time series are present, one ﬁnal re-

quirement must be fulﬁlled to know whether we can obtain a

dimension and whether the complete spectrum of Lyapunov

exponents (another method to visualize chaos) still needs to

be employed.

Eckmann and Ruelle (1992) discuss the size of the data set

required to estimate Lyapunov dimensions and exponents.

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296 B. Rojo-Garibaldi et al.: Nonlinear analysis of the occurrence of hurricanes

Figure 5. Lyapunov exponent with m=4, τ=9 and m=5, τ=10, with the Kantz method (a). Lyapunov exponent with the same values

with the Rosenstein method (b).

Figure 6. The correlation dimension D2corresponding to the occurrence of hurricanes in the years 1749–2012 in the Gulf of Mexico and

the Caribbean Sea. Curves for different dimensions of the attractor (yaxis) (a). Same information for the D2with a logarithmic scale on the

xaxis (b),

When these dimensions and exponents measure the diver-

gence rate with near-initial conditions, they require a num-

ber of neighbors for a given reference point. These neighbors

may be within a sphere of radius (r) and of a given diameter

(d) of the reconstructed attractor.

We then have the requirements for the Eckmann and Ru-

elle (1992) condition to obtain the Lyapunov exponents as

logN > D log1

ρ,(3)

where Dis the dimension of the attractor, Nis the number

of data points and r

d=ρ. For ρ=0.1 in Eq. (3), Nmay be

chosen such that

N > 10D.(4)

Our time series met this requirement; therefore, it supports

our previous results.

The attractor dimension was mainly obtained because this

value tells us the number of parameters or degrees of freedom

necessary to control or understand the temporal evolution of

our system in the phase space and helps us to determine how

chaotic our system is. Using the previous methods, a ﬁnal

fractal dimension of D2=2.2 was obtained. Following the

embedding laws, it stands to reason that m>D2 (Sauer and

Yorke, 1993; Kantz and Schreiber, 2004; Bradley and Kantz,

2015). The criterion of Ruelle (1990) was used to corrobo-

rate that the obtained dimension of the attractor is reliable,

where N=10

D2

2; once the data fulﬁll this requirement, we

can say that the dimension of the attractor is reliable. Finally,

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B. Rojo-Garibaldi et al.: Nonlinear analysis of the occurrence of hurricanes 297

Figure 7. The iterated functions system (IFS) test applied to the

time series of the number of hurricanes that occurred in the Gulf of

Mexico and the Caribbean Sea between the years 1749 and 2012.

the results indicated that at least three parameters are needed

to characterize our system, since the 2.2 dimension indicates

that the attractor dimension falls between two and three.

The spectrum of the Lyapunov exponent gives 0.09983,

−0.07443, −0.23387 and −0.73958; therefore, the total sum

is λi= −0.9480, and according to the previous theory, it is

enough have at least one positive exponent in the spectrum

of our system in order to have chaotic behavior. Finally, the

total sum of the spectrum of Lyapunov exponents was neg-

ative, indicating that there is a stable attractor, as mentioned

previously. However, since the stable attractor was not easily

distinguished, we used a ﬁnal method in order to conﬁrm if

our system presented some chaotic dynamic behavior. This

method comprised the iterated functions system test (IFS)

(Fig. 7).

Using Fig. 7, it can be observed that the points represent-

ing our system occupy the entire space; according to the IFS

test, there are two possible explanations:

1. The distribution belongs to a white noise signal and in

systems without experimental noise, the point distribu-

tion gives a single curve (Jensen et al., 1985). However,

the previous Hurst exponent obtained was not equal to

zero; therefore, the white noise was also discarded with

the autocorrelation function.

2. The system is chaotic with high dimensionality. So far,

our results have converged on the occurrence of hurri-

canes in the Gulf of Mexico and the Caribbean Sea be-

ing a chaotic system, so it is feasible to adopt the second

explanation. Conversely however, our Lyapunov expo-

nent ﬁgure was not ﬂat and it did not seem to ﬂatten

as the dimension of embedding increased, which, ac-

cording to Rosenstein et al. (1993), would mean that

our system is not chaotic; although the Lyapunov ex-

ponent increased with the decrease in the embedment

dimension, which is, again, a characteristic of chaotic

systems. It was then also possible to obtain a dimension

of the attractor and a positive Lyapunov exponent.

Our results were not easy to interpret because the series pre-

sented certain periodic characteristics in an oscillatory fash-

ion and simultaneously showed chaotic behavior. According

to Rojo-Garibaldi et al. (2016), the series of hurricanes which

had spectral analyzes carried out presented strong periodici-

ties that correspond to sunspots, which are believed to have

caused the periodic behavior mentioned above. According to

Zeng et al. (1990), the spectral power analysis is often used

to distinguish a chaotic or quasi-periodic behavior of peri-

odic structures and to identify different periods embedded

in a chaotic signal. Although, as Schuster (1988) and Tso-

nis (1992) mention, the power spectrum is not only char-

acteristic of a process of deterministic chaos but also of a

linear stochastic process. In our case, this behavior was not

observed in the spectra obtained, which allowed us to de-

tect periodic signals. The spectra identify two types of be-

havior in our system. On one hand, there are periodic be-

haviors associated with external forcing, such as the sunspot

cycle, giving the system sufﬁcient order to develop; whilst

on the other hand, external forcing presents a chaotic behav-

ior, which gives the system a certain disorder and allows it

to be able to adapt to new changes and evolve. The IFS test

showed that the occurrence of hurricanes in the Gulf of Mex-

ico and the Caribbean Sea is chaotic with high dimension-

ality. Fraedrich and Leslie (1989) analyzed the trajectories

of cyclones in the region of Australia and calculated the di-

mensionality of this process, obtaining a result of between

six and eight, i.e., a chaotic process of high dimensionality,

which is similar to what we ﬁnd with the IFS method. Halsey

and Jensen (2004) furthermore postulate that hurricanes con-

tain a large number of dimensions in phase space.

One possible explanation is localized within a boundary

where chaos and order are separated; this boundary is com-

monly known as the ”edge of chaos” (Langton, 1990; Mira-

montes et al., 2001). Miramontes et al. (2001) found this type

of behavior in ants of the genus Leptothorax, when studying

them individually and in groups. In the former, the behavior

was periodic, whilst in the latter, the behavior was chaotic.

In our case, we believe that the chaotic behavior is due to the

individual behavior or the hurricane category, as the high di-

mensionality suggested by the IFS test agrees with the high

dimensionality reported by Fraedrich and Leslie (1989) ob-

tained by studying the trajectories of cyclones – that is, by

studying them individually – while the periodic response is

due to the behavior of hurricanes as a whole.

Finally, an entropy test was performed using non-linear

methods and locally linear prediction (making the predic-

tion at one step), with both methods showing a predictability

value of 2.78 years. The locally linear prediction method was

applied as follows: the last known state of the system, repre-

sented by a vector x=[x(n),x(n+τ ),...,x(n+(m−1) τ)],

was determined, where mis the embedment dimension and

τis the delay time. We then found the pnearby states (usu-

ally close neighbors of x) of the system which represent what

has happened in the past, these were obtained by calculating

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298 B. Rojo-Garibaldi et al.: Nonlinear analysis of the occurrence of hurricanes

Figure 8. Prediction of the hurricanes number in the Gulf of Mexico

and the Caribbean Sea by means of non-linear methods, and the en-

tropy test. The solid black line represents the number of hurricanes

observed from 2013 to 2017. The black points are the prediction

and the triangles are the error in the prediction, considered as the

observed values minus the predicted values.

their distances to x. The idea is then to adjust a map that ex-

trapolates xand its neighboring pto determine the following

values (Dasan et al., 2002). Based on the above, the value of

the embedding dimension and the delay time were changed.

Different values of mwere used in order to elucidate the most

accurate result; this was obtained with a dimension of m=4

and τ=9, which are the values for which the attractor of the

system was obtained. Therefore, a good prediction is possible

until t=t0+3 (Fig. 8). Furthermore, if we get away from the

measured data (reported hurricanes) the uncertainty grows in

an oscillatory way. For the ﬁrst two data (2013 and 2014),

the absolute error in the prediction (observed value – pre-

dicted) is less than 0.2, for the third and fourth value (2015

and 2016) it is between 0.2 and 0.3 and for 2017 the error is

much greater and gives an overestimation of one hurricane

(Fig. 8). An important result of this study is that it allows one

to establish the predictability range of a system.

4 Conclusions

The results obtained with the nonlinear analysis suggested

a chaotic behavior in our system, mainly based on the Lya-

punov exponents and correlation dimension, among others.

However, the Hurst exponent indicated that our system did

not follow a chaotic behavior. In order to be able to corrobo-

rate our results, we employed the IFS method, which led us

to believe that the hurricane time series in the Gulf of Mexico

and the Caribbean Sea from 1749 to 2012 had a chaotic edge.

It is important to emphasize that this study was prepared as

an attempt at understanding the behavior of the occurrence of

hurricanes from a historical perspective, as this type of phe-

nomenon is part of an ocean–atmosphere interaction that has

been changing over time, hence the value of our contribu-

tion. However, we are aware that from the time the study was

conducted to the present date there are new records, which

will make it possible to carry out new studies and apply new

methods.

Data availability. The data are available upon request from the au-

thor.

Author contributions. All the authors contributed equally to this

work.

Competing interests. The authors declare that they have no conﬂict

of interest.

Acknowledgements. This work was ﬁnancially supported by the

Instituto de Ciencias del Mar y Limnología de la Universidad

Nacional Autónoma de México, projects 144 and 145. BR-G is

grateful for the CONACYT scholarship that supported her study

at the Posgrado en Ciencias del Mar y Limnología, Universidad

Nacional Autónoma de México.

Edited by: Vicente Perez-Munuzuri

Reviewed by: three anonymous referees

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