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Hurricanes in the Gulf of Mexico and the Caribbean Sea and their relationship with sunspots

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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
StSj
is calculated iterating both points, given by the
following:
Ri=Si+1Sj+1
SlSJ
.(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+(m1) 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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