Hurricanes are complex systems that carry large amounts of energy. Their impact often produces natural disasters 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 construction 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 information. 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 category or individually regardless of their category and their behavior on a regular basis.

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 trajectory and intensity once tracking has begun, as well as their number and intensity from one year to the next. However, 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 information. However, we decided to make a different contribution by carrying out a nonlinear analysis of a time series of hurricanes 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 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, including 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; Gholami et al. (2015) simulate groundwater levels using dendrochronology and an artificial neural network model for the southern Caspian coast in Iran. Furthermore, theories of deterministic chaos and fractal structure have already been applied to atmospheric boundary data (Tsonis and Elsner, 1988; Zeng et al., 1992), e.g., to the pulse of severe rain time series (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 contexts; 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 embedded parameters and the exponential law for the growth of distances can be explicitly tested (Rigney et al., 1993; Rosenstein et al., 1993). This exponent provides a qualitative characterization of the dynamic behavior and the predictability measurement (Atari et al., 2003). The algorithms usually employed 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 fluctuations 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 provide a strong test of whether the calculated numbers can actually be interpreted as Lyapunov exponents of a deterministic 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 implement and robust to changes in the following quantities: embedded 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, which in turn will help to reinforce prevention measures for this type of hydrometeorological phenomenon.

A detailed analysis of historical reports was carried out in order to obtain the annual time series of hurricane occurrence, 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 sailing 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 currently 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 impossible with the records available in other regions.

Hurricanes between 1749 and 2012. The dashed line shows the linear trend (after Rojo-Garibaldi et al., 2016).

Historical hurricanes were included only if they were reported in two or more databases and met both of the following criteria: the reported hurricanes that touched land and those that remained in the ocean; on the other hand, the followed hurricanes were studied considering their average duration and their maximum time (9 and 19 days, respectively). This was done in order to avoid counting more than one specific hurricane reported in different places within a short period time; to do this, we followed the proposed method by Rojo-Garibaldi et al. (2016).

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

Before performing the nonlinear analysis of the time series, we removed the
trend; thus, the series was prepared according 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:

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 autocorrelation in a data series.

The Lyapunov exponent is invariant under soft transformations, because it
describes long-term behavior, providing an objective characterization of the
corresponding dynamics (Kantz and Schreiber, 2004). The presence of chaos in
dynamic systems can be solved using this exponent, as it quantifies the
exponential convergence or divergence of initially close trajectories in the
state space and estimates the amount of chaos in a system (Rosenstein et al.,
1993; Haken, 1981; Wolf, 1986). The Lyapunov exponent (

The iterated function analysis (IFS) is an easier and simpler 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 system attractor was a fractal type or not. It is important to compute the Lyapunov exponent, so we used the algorithms proposed by Kantz (1994) and Rosenstein et al. (1993) to do so.

The mutual information method

Figure 1 shows the evolution of the number of hurricanes from 1749 to 2012
and the linear trend. To have a qualitative 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 ”delay
method” (Fig. 2). This was also used to elucidate the time lag for an
optimal embedding in the data set. The optimal time lag (

The most robust method to identify chaos within the system is the Lyapunov
exponent. Prior to obtaining the exponent, it was necessary to calculate the
time lag and the embedding dimension, and for the latter, the Theiler window
was used. The time lag was obtained via three different methods:

The method of constructing delays, which is observed visually in Fig. 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

The autocorrelation function method, which is based solely on linear statistics (Fig. 3).

the first zero of the function, and

the moment in which the
autocorrelation function decays as

False close neighbors with a time lag of 10, where the embedding 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.

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.

Lyapunov exponent with

The correlation dimension

It was possible to observe the difference in the time lag obtained through
the autocorrelation function and the mutual information; however, it is
necessary to use only one result. Through the space-time separation graphic
and the false close neighbors method, we obtained embedding dimensions of

The idea of the false close neighbors algorithm is that at each point in the
time series,

The Lyapunov (

There was a difference between placing the attractor in an embedding
dimension of

It was possible to obtain the correlation dimension

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 right panel on Fig. 7 shows the slope trend of the majority of the slopes
of the correlation integral (

Even when all the requirements necessary to apply the nonlinear analysis to our time series are present, one final requirement 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. When these dimensions and
exponents measure the divergence rate with near-initial conditions, they
require a number of neighbors for a given reference point. These neighbors
may be within a sphere of radius (

We then have the requirements for the Eckmann and Ruelle (1992)
condition to obtain the Lyapunov exponents as

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

The spectrum of the Lyapunov exponent gives 0.09983,

Using Fig. 7, it can be observed that the points representing our system
occupy the entire space; according to the IFS test, there are two possible
explanations:

The distribution belongs to a white noise signal and in systems without experimental noise, the point distribution 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.

The system is chaotic with high dimensionality. So far, our results have converged on the occurrence of hurricanes in the Gulf of Mexico and the Caribbean Sea being a chaotic system, so it is feasible to adopt the second explanation. Conversely however, our Lyapunov exponent figure was not flat and it did not seem to flatten as the dimension of embedding increased, which, according to Rosenstein et al. (1993), would mean that our system is not chaotic; although the Lyapunov exponent 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.

One possible explanation is localized within a boundary where chaos and
order are separated; this boundary is commonly known as the ”edge of
chaos” (Langton, 1990; Miramontes et al., 2001). Miramontes et al. (2001)
found this type of behavior in ants of the genus

Finally, an entropy test was performed using non-linear
methods and locally linear prediction (making the prediction 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, represented by a vector

Prediction of the hurricanes number in the Gulf of Mexico and the Caribbean Sea by means of non-linear methods, and the entropy 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.

The results obtained with the nonlinear analysis suggested a chaotic behavior in our system, mainly based on the Lyapunov 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 corroborate 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 phenomenon is part of an ocean–atmosphere interaction that has been changing over time, hence the value of our contribution. 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.

The data are available upon request from the author.

All the authors contributed equally to this work.

The authors declare that they have no conflict of interest.

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