the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Magnetospheric chaos and dynamical complexity response during storm time disturbance

### Irewola Aaron Oludehinwa

### Olasunkanmi Isaac Olusola

### Olawale Segun Bolaji

### Olumide Olayinka Odeyemi

### Abdullahi Ndzi Njah

In this study, we examine the magnetospheric chaos and dynamical complexity
response to the disturbance storm time (*D*_{st}) and solar wind electric
field (VB_{s}) during different categories of geomagnetic storm (minor, moderate and major geomagnetic storm). The time series data of the *D*_{st}
and VB_{s} are analysed for a period of 9 years using non-linear
dynamics tools (maximal Lyapunov exponent, MLE; approximate entropy, ApEn;
and delay vector variance, DVV). We found a significant trend between each
non-linear parameter and the categories of geomagnetic storm. The MLE and
ApEn values of the *D*_{st} indicate that chaotic and dynamical complexity
responses are high during minor geomagnetic storms, reduce at moderate
geomagnetic storms and decline further during major geomagnetic storms.
However, the MLE and ApEn values obtained from VB_{s} indicate that
chaotic and dynamical complexity responses are high with no significant
difference between the periods that are associated with minor, moderate and
major geomagnetic storms. The test for non-linearity in the *D*_{st} time
series during major geomagnetic storm reveals the strongest non-linearity
features. Based on these findings, the dynamical features obtained in the
VB_{s} as input and *D*_{st} as output of the magnetospheric system
suggest that the magnetospheric dynamics are non-linear, and the solar wind
dynamics are consistently stochastic in nature.

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The response of chaos and dynamical complexity behaviour with respect to
magnetospheric dynamics varies (Tsurutani et al., 1990). This is due to
changes in the interplanetary electric fields imposed on the magnetopause
and those penetrating the inner magnetosphere and sustaining convection,
thereby initiating a geomagnetic storm (Dungey, 1961; Pavlos et al., 1992). A
prolonged southward turning of interplanetary magnetic field (IMF, *B*_{z}),
which indicates that solar-wind–magnetosphere coupling is in progress, was
confirmed on many occasions for which such a geomagnetic storm was driven by
co-rotating interaction regions (CIRs), by the sheath preceding an
interplanetary coronal mass ejection (ICME), or by a combination of the
sheath and an ICME magnetic cloud (Gonzalez and Tsurutani, 1987; Tsurutani
and Gonzalez, 1987; Tsurutani et al., 1988; Cowley, 1995; Tsutomu, 2002;
Yurchyshyn et al., 2004; Kozyra et al., 2006; Echer et al., 2008; Meng et
al., 2019; Tsurutani et al., 2020; Echer et al., 2006). The sporadic magnetic reconnection
between the southward component of the Alfvén waves and the earth's
magnetopause leads to isolated substorms or convection events such as the high-intensity long-duration continuous AE activity (HILDCAA, where AE represents auroral
electrojet) which are shown to
last from days to weeks (Akasofu, 1964; Tsurutani and Meng, 1972; Meng et al.,
1973; Tsurutani and Gonzalez, 1987; Hajra et al., 2013; Liou et al., 2013;
Mendes et al., 2017; Hajra and Tsurutani, 2018; Tsurutani and Hajra, 2021; Russell, 2001).
Notably, the introduction of the disturbance storm time (*D*_{st}) index
(Sugiura, 1964; Sugiura and Kamei, 1991) unveiled a quantitative measure
of the total energy of the ring current particles. Therefore, the *D*_{st}
index remains one of the most popular global indicators that can precisely
reveal the severity of a geomagnetic storm (Dessler and Parker, 1959).

The *D*_{st} fluctuations exhibit different signatures for different
categories of geomagnetic storm. Ordinarily, one can easily anticipate that
fluctuations in a *D*_{st} signal appear chaotic and complex. These may
arise from the changes in the interplanetary electric fields driven by the
solar-wind–magnetospheric coupling processes. At different categories of
geomagnetic storm, fluctuations in the *D*_{st} signals differ (Oludehinwa
et al., 2018). One obvious reason is that as the intensity of the
geomagnetic storm increases, the fluctuation behaviour in the *D*_{st}
signal becomes more complex and non-linear in nature. It has been established
that the electrodynamic response of the magnetosphere to solar wind drivers
are non-autonomous in nature (Price and Prichard, 1993; Price et al., 1994;
Johnson and Wings, 2005). Therefore, the chaotic analysis of the
magnetospheric time series must be related to the concept of input–output
dynamical process (Russell et al., 1974; Burton et al., 1975; Gonzalez et
al., 1989, 1994). Consequently, it is necessary to examine
the chaotic behaviour of the solar wind electric field (VB_{s}) as input
signals and the magnetospheric activity index (*D*_{st}) as output during
different categories of geomagnetic storms.

Several works have been presented on the chaotic and dynamical complexity
behaviour of the magnetospheric dynamics based on an autonomous concept, i.e.
using the time series data of magnetospheric activity alone such as auroral
electrojet (AE), amplitude lower (AL) and *D*_{st} index (Vassiliadis et
al., 1990; Baker and Klimas, 1990; Vassiliadis et al., 1991; Shan et al.,
1991; Pavlos, 1994; Klimas et al., 1996; Valdivia et al., 2005;
Mendes et al., 2017; Consolini, 2018). Authors found evidence of
low-dimensional chaos in the magnetospheric dynamics. For instance, the
report by Vassiliadis et al. (1991) shows that the computation of Lyapunov
exponent for AL index time series gives a positive value of Lyapunov
exponent, indicating the presence of chaos in the magnetospheric dynamics.
Unnikrishnan (2008) studied the deterministic chaotic behaviour in the
magnetospheric dynamics under various physical conditions using AE index
time series and found that the seasonal mean value of Lyapunov exponent in
winter season during quiet periods (0.7±0.11 min^{−1}) is higher
than that of the stormy periods (0.36±0.09 min^{−1}). Balasis et
al. (2006) examined the magnetospheric dynamics in the *D*_{st} index time
series from pre-magnetic storm to magnetic storm period using fractal
dynamics. They found that the transition from anti-persistent to persistent
behaviour indicates that the occurrence of an intense geomagnetic storm is
imminent. Balasis et al. (2009) further reveal the dynamical complexity
behaviour in the magnetospheric dynamics using various entropy measures.
They reported a significant decrease in dynamical complexity and an
accession of persistency in the *D*_{st} time series as the magnetic storm
approaches. Recently, Oludehinwa et al. (2018) examined the non-linearity
effects in *D*_{st} signals during minor, moderate and major geomagnetic
storms using recurrence plots and recurrence quantification analyses. They
found that the dynamics of the *D*_{st} signal is stochastic during minor
geomagnetic storm periods and deterministic as the geomagnetic storm
increases.

Also, studies describing the solar wind and magnetosphere as a
non-autonomous system have been extensively investigated. Price et al. (1994) examine the non-linear input–output analysis of AL index and different
combinations of interplanetary magnetic field (IMF) with solar wind
parameters as input functions. They found that only a few of the input
combinations show any evidence whatsoever for non-linear coupling between the
input and output for the interval investigated. Pavlos et al. (1999)
presented further evidence of magnetospheric chaos. They compared the
observational behaviour of the magnetospheric system with the results
obtained by analysing different types of stochastic and deterministic
input–output systems and asserted that a low-dimensional chaos is evident in
magnetospheric dynamics. Prabin Devi et al. (2013) studied the magnetospheric
dynamics using AL index and the southward component of IMF (*B*_{z}). They
observed that the magnetosphere and turbulent solar wind have values
corresponding to non-linear dynamical system with chaotic behaviour. The
modelling and forecasting approach have been applied to magnetospheric time
series using non-linear models (Valdivia et al., 1996; Vassiliadis et al.,
1999; Vassiliadis, 2006; Balikhin et al., 2010). These efforts have improved
our understanding that the concept of non-linear dynamics can reveal some
hidden dynamical information in the observational time series. In addition
to these non-linear effects in *D*_{st} signals, a measure of the exponential
divergence and convergence within the trajectories of a phase space known as
maximal Lyapunov exponent (MLE), which has the potential to depict the
chaotic behaviour in the *D*_{st} and VB_{s} time series during a minor,
moderate or major geomagnetic storm, has not been investigated. In
addition, to the best of our knowledge, computation of approximate entropy
(ApEn) that depicts the dynamical complexity behaviour during different
categories of geomagnetic storm has not been reported in the literature. The
test for non-linearity through delay vector variance (DVV) analysis that
reveals the non-linearity features in *D*_{st} and VB_{s} time series
during minor, moderate and major geomagnetic storms is not well known. It is
worthy to note that understanding the dynamical characteristics in the
*D*_{st} and VB_{s} signals at different categories of geomagnetic storms
will provide useful diagnostic information to different conditions of space
weather phenomenon. Consequently, this study attempts to carry out
comprehensive numerical analyses to unfold the chaotic and dynamical
complexity behaviour in the *D*_{st} and VB_{s} signals during minor,
moderate and major geomagnetic storms. In Sect. 2, our methods of data
acquisition are described. Also, the non-linear analysis that we employed in
this investigation is detailed. In Sect. 3, we unveil our results and
engage in the discussion of results in Sect. 4.

The *D*_{st} index is derived by measurements from ground-based magnetic
stations at low-latitude observatories around the world and depicts mainly
the variation of the ring current, as well as the Chapman–Ferraro
magnetopause currents, and tail currents to a lesser extent (Sugiura, 1964;
Feldstein et al., 2005, 2006; Love and Gannon, 2009). Due
to its global nature, the *D*_{st} time series provides a measure of how intense
a geomagnetic storm was (Dessler and Parker, 1959). In this study, we
considered *D*_{st} data for the period of 9 years from January to
December between 2008 and 2016 which were downloaded from the World Data
Centre for Geomagnetism, Kyoto, Japan (http://wdc.kugi.kyoto-u.ac.jp/dst_final/index.html, last access: 15 May 2021). We use the classification
of geomagnetic storms as proposed by Gonzalez et al. (1994) such that
*D*_{st} index value in the ranges $\mathrm{0}\le {D}_{\mathrm{st}}\le -\mathrm{50}$ nT, $-\mathrm{50}\le {D}_{\mathrm{st}}\le -\mathrm{100}$ nT and $-\mathrm{100}\le {D}_{\mathrm{st}}\le -\mathrm{250}$ nT are classified as minor, moderate and
major geomagnetic storms, respectively, and each time series is being
classified based on its minimum *D*_{st} value. The solar wind electric
field (VB_{s}) data are archived from the National Aeronautics and Space
Administration, Space Physics Facility (https://omniweb.gsfc.nasa.gov/form/dx1.html, last access: 15 May 2021). The sampling time of *D*_{st} and
VB_{s} time series data was 1 h. It is well known that the dynamics of the solar wind contribute to the driving of the magnetosphere (Burton et al., 1975). Furthermore, we took the solar wind electric field (VB_{s}) as
the input signal (Price and Prichard, 1993; Price et al., 1994). The
VB_{s} was categorized according to the periods of minor, moderate and
major geomagnetic storms. Then, the *D*_{st} and VB_{s} time series were
subjected to a variety of non-linear analytical tools explained as follows.

## 2.1 Phase space reconstruction and observational time series

An observational time series can be defined as a sequence of scalar measurements of some quantity, which is a function of the current state of the system taken at multiples of a fixed sampling time. In non-linear dynamics, the first step in analysing an observational time series is to reconstruct an appropriate state space of the system. Takens (1981) and Mañé (1981) stated that one time series or a few simultaneous time series are converted to a sequence of vectors. This reconstructed phase space has all the dynamical characteristic of the real phase space provided the time delay and embedding dimension are properly specified.

where *X*(*t*) is the reconstructed phase space, *x*(*t*) is the original time
series data, *τ* is the time delay and *m* is the embedding dimension.
An appropriate choice of *τ* and *m* is needed for the reconstruction
of phase space, which is determined by average mutual information and false
nearest neighbour, respectively.

## 2.2 Average mutual information (AMI)

The method of average mutual information (AMI) is one of the non-linear
techniques used to determine the optimal time delay (*τ*) required for
phase space reconstruction in observational time series. The time delay
mutual information was proposed by Fraser and Swinney (1986) instead of an
autocorrelation function. This method takes into account non-linear
correlations within the time series data. It measures how much information
can be predicted about one time series point, given full information about
the other. For instance, the mutual information between *x*_{i} and
*x*_{(i+τ)} quantifies the information in state *x*_{(i+τ)} under
the assumption that information at the state *x*_{i} is known. The AMI for a
time series *x*(*t*_{i}), $i=\mathrm{1},\mathrm{2},\mathrm{\dots},N$ is
calculated as

where *x*(*t*_{i}) is the *i*th element of the time series, *T*=*k*Δ*t*
($k=\mathrm{1},\mathrm{2},\mathrm{\dots},{k}_{max}$), *P*(*x*(*t*_{i})) is the
probability density at *x*(*t*_{i}), and $P\left(x\right({t}_{i}),x({t}_{i}+T\left)\right)$ is the joint probability density at the pair *x**t*(*t*_{i}) and *x*(*t*_{i}+*T*). The time delay (*τ*) of the first minimum of
AMI is chosen as optimal time delay (Fraser and Swinney, 1986; Fraser, 1986). Therefore,
the AMI was applied to the *D*_{st} and VB_{s} time series, and the plot of
AMI versus time delay is shown in Fig. 3. We notice that the AMI showed
the first local minimum at roughly *τ*=15 h. Furthermore, the values of
*τ* near this value of ∼ 15 h maintain constancy for both
VB_{s} and *D*_{st}. In the analysis, *τ*=15 h was used as the
optimal time delay for the computation of maximal Lyapunov exponent.

## 2.3 False nearest neighbour (FNN)

In determining the optimal choice of embedding dimension (*m*), the false
nearest neighbour method was used in the study. The method was suggested by
Kennel et al. (1992). The concept is based on how the number of neighbours of
a point along a signal trajectory changes with increasing embedding
dimension. With increasing embedding dimension, the false neighbour will no
longer be neighbours; therefore, by examining how the number of neighbours
changes as a function of dimension, an appropriate embedding dimension can
be determined. For instance, suppose we have a one-dimensional time series.
We can construct a time series *y*(*t*) of *D*-dimensional points from the
original one-dimensional time series *x*(*t*) as follows:

where *τ* and *D* are time delay and embedding dimension. Using the
formula from Kennel et al. (1992) and Wallot and Monster (2018), if we have a
*D*-dimensional phase space and denote the *r*th nearest neighbour of a
coordinate vector ** y**(

*t*) by

*y*

^{(r)}(

*t*), then the square of the Euclidean distance between

*y*(

*t*) and the

*r*th nearest neighbour is

Now applying the logic outlined above, we can go from a *D*-dimensional
phase space to *D*+1-dimensional phase space by time-delay embedding,
adding a new coordinate to *y*(*t*), and ask what is the squared distance
between *y*(*t*) and the same *r*th nearest neighbour:

As explained above, if the one-dimensional time series is already properly
embedded in *D* dimensions, then the distance *R* between *y*(*t*) and the
*r*th nearest neighbour should not change appreciably by some distance
criterion *R*_{tol} (i.e. *R*<*R*_{tol}). Moreover, the distance of the nearest
neighbour when embedded into the next higher dimension relative to the size
of the attractor should be less than some criterion *A*_{tol} (i.e.
${R}_{D+\mathrm{1}}<{A}_{\mathrm{tol}}$). Doing this for the nearest neighbour of each coordinate
will result in many false nearest neighbours when embedding is insufficient
or in few (or no) false neighbours when embedding is sufficient. In the
analysis, the FNN was applied to the *D*_{st} and VB_{s} time series to
detect the optimal value of embedding dimension (*m*). Figure 4 shows a
sample plot of the percentage of false nearest neighbour against embedding
dimension in one of the months under investigation (other months show
similar results; thus, for brevity we depict only one of the results). We
notice that the false nearest neighbour attains its minimum value at *m*≥5,
indicating that embedding dimension (*m*) values from *m*≥5 are optimal.
Therefore, *m*=5 was used for the computation of maximal Lyapunov exponent.

## 2.4 Maximal Lyapunov exponent (MLE)

The maximal Lyapunov exponent (MLE) is one of the most popular non-linear
dynamics tools used for detecting chaotic behaviour in a time series. It
describes how small changes in the state of a system grow at an exponential
rate and eventually dominate the behaviour. An important indication of
chaotic neighbour of a dissipative deterministic system is the existence of a
positive Lyapunov exponent. A positive MLE signifies divergence of
trajectories in one direction or expansion of an initial volume in this
direction. On the other hand, a negative MLE exponent implies convergence of
trajectories or contraction of volume along another direction. The algorithm
proposed by Wolf et al. (1985) for estimating MLE is employed to compute the
chaotic neighbour of the *D*_{st} and VB_{s} time series at minor, moderate
and major geomagnetic storms. Other methods of determining MLE include
Rosenstein's method, Kantz's method and so on. In this study, the MLE for
minor, moderate and major geomagnetic storm periods was computed with *m*=5
and *τ*=15 h as shown in Figs. 5 and 6 (bar plots) for *D*_{st} and
VB_{s}. The calculation of MLE is explained as follows: given a sequence
of vector ** x**(

*t*), an

*m*-dimensional phase space is formed from the observational time series through an embedding theorem as

where *m* and *τ* are as defined earlier; after reconstructing the
observational time series, the algorithm locates the nearest neighbour (in
Euclidean sense) to the initial point $x\left({t}_{\mathrm{0}}\right),\mathrm{\dots},x({t}_{\mathrm{0}}+(m-\mathrm{1}\left)\mathit{\tau}\right)$ and denotes the distance between these two
points *L*(*t*_{0}). At a later point *t*_{1}, the initial length will have
evolved to length *L*^{′}(*t*_{1}). Then the MLE is calculated as

where *M* is the total number of replacement steps. We look for a new data point
that satisfies two criteria reasonably well: its separation, *L*(*t*_{1}),
from the evolved fiducial point is small. If an adequate replacement point
cannot be found, we retain the points that were being used. This procedure
is repeated until the fiducial trajectory has traversed the entire data.

## 2.5 Approximate entropy (ApEn)

Approximate entropy (ApEn) is one of the non-linear dynamics tools that
measures the dynamical complexity in observational time series. The concept
was proposed by Pincus (1991), who provide a generalized measure of
regularity, such that it accounts for the logarithm likelihood in the
observational time series. For instance, a dataset of length *N* that
repeat itself for *m* points within a boundary will again repeat itself for
*m*+1 points. Because of its computational advantage, ApEn has been widely
used in many disciplines to study dynamical complexity (Pincus and
Kalman, 2004; Pincus and Goldberger, 1994; McKinley et al., 2011;
Kannathan et al., 2005; Balasis et al., 2009; Shujuan and Weidong, 2010;
Moore and Marchant, 2017). The ApEn is computed using the formula below:

where ${C}_{i}^{m}\left(r\right)=\frac{\mathrm{1}}{N-m+\mathrm{1}}\sum _{j=\mathrm{1}}^{N-m+\mathrm{1}}\mathrm{\Theta}\left(r-\u2225{x}_{i}-{x}_{j}\u2225\right)$ is the
correlation integral, *m* is the embedding dimension and *r* is the
tolerance. To compute the ApEn for the *D*_{st} and VB_{s} time series
classified as minor, moderate and major geomagnetic storms from 2008 to 2016,
we choose *m*=3 and *τ*=1 h. We refer interested readers to the works of Pincus (1991),
Kannathal et al. (2005) and Balasis et al. (2009) where all the computational steps regarding ApEn were explained in detail.
Figures. 5 and 6 depict the stem plot of ApEn for *D*_{st} and VB_{s}
from 2008 to 2016.

## 2.6 Delay vector variance (DVV) analysis

The delay vector variance (DVV) is a unified approach in analysing and
testing for non-linearity in a time series (Gautama et al., 2004; Mandic et
al., 2007). The basic idea of the DVV is that if two delay vectors of a
predictable signal are close to each other in terms of the Euclidean
distance, they should have a similar target. For instance, when a time delay
(*τ*) is embedded into a time series *x*(*k*), $k=\mathrm{1},\mathrm{2},\mathrm{\dots},N$, then a reconstructed phase space vector is formed
which represents a set of delay vectors (DVs) of a given dimension.

Reconstructing the phase space, a set (*λ*_{k}) is generated by
grouping those DVs that are within a certain Euclidean distance to DVs (*X*(*k*)). For a given embedding dimension (*m*), a measure
of unpredictability *σ*^{∗2} is computed over all pairwise
Euclidean distances between delay vector as

Then, sets *λ*_{k}(*r*_{d}) are generated as the sets which consist of
all delay vectors that lie closer to *x*(*k*) than a certain distance *r*_{d}.

For every set *λ*_{k}(*r*_{d}), the variance of the corresponding
target *σ*^{∗2}(*r*_{d}) is

where *σ*^{∗2}(*r*_{d}) is target variance against the
standardized distance, indicating that Euclidean distance will be varied in a
manner standardized with respect to the distribution of pairwise distance
between DVs. The iterative amplitude-adjusted Fourier transform (IAAFT) method
is used to generate the surrogate time series (Kugiumtzis, 1999). If the
surrogate time series yields a DV plots similar to the original time series
and the scattered plot coincides with the bisector line, then the original
time series can be regarded as linear (Theiler et al., 1992; Gautama et al.,
2004; Imitaz, 2010; Jaksic et al., 2016). On the other hand, if the
surrogate time series yields DV plot that is not similar to that of the
original time series, then the deviation from the bisector lines indicates
non-linearity. The deviation from the bisector lines grows as a result of the
degree of non-linearity in the observational time series.

where ${\mathit{\sigma}}_{\mathrm{s},\mathrm{i}}^{\ast \mathrm{2}}({r}_{\mathrm{d}}$) is the target variance at the
span *r*_{d} for the *i*th surrogate. To carry out the test for
non-linearity in the *D*_{st} signals (*m*=3, *n*_{d}=3), the number of
reference DVs = 200 and the number of surrogates *N*_{s}=25 were used in all of the
analysis. Then we examined the non-linearity response at minor, moderate and major geomagnetic storms.

In this study, *D*_{st} and VB_{s} time series from January to December
were analysed for the period of 9 years (2008 to 2016) to examine the
chaotic and dynamical complexity response in the magnetospheric dynamics
during minor, moderate and major geomagnetic storms. Figures 1 and 2
display the samples of fluctuation signatures of *D*_{st} and VB_{s}
signals classified as (a) minor, (b) moderate and (c) major geomagnetic
storms. The plot of average mutual information against time delay (*τ*)
shown in Fig. 3 depicts that the first local minimum of the AMI function
was found to be roughly at *τ*=15 h. Furthermore, we notice that the
values of *τ* near this value of ∼ 15 h maintain
constancy for both VB_{s} and *D*_{st}. Also, in Fig. 4, we display
the plot of the percentage of false nearest neighbour against embedding
dimension (*m*). It is obvious that a decrease in false nearest neighbour
when increasing the embedding dimension drops steeply to zero at the optimal
dimension (*m*=5); thereafter, the false neighbours stabilize at that *m*=5
for VB_{s} and *D*_{st}. Therefore, *m*=5 and *τ*=15 h were used for
the computation of MLE at different categories of geomagnetic storm, while
*m*=3 and *τ*=1 h are applied for the computation of ApEn values.

The results of MLE (bar plot) and ApEn (stem plot) for *D*_{st} at minor,
moderate and major geomagnetic storms are shown in Fig. 5. During minor
geomagnetic storms, we notice that the value of MLE ranges between 0.07 and
0.14 for most of the months classified as minor geomagnetic storms.
Similarly, the ApEn (stem plot) ranges between 0.59 and 0.83. It is obvious
that strong chaotic behaviour with high dynamical complexity are associated
with minor geomagnetic storms. During moderate geomagnetic storms, (see Fig. 5b), we observe a reduction in MLE values (0.04–0.07)
compared to minor geomagnetic storm periods. Within the observed values of
MLE during moderate geomagnetic storms, we found a slight rise of MLE in the
following months: March in 2008; April in 2011; January, February, and April in 2012;
July, August, September, October, and November in 2015; and November in 2016. Also,
the ApEn revealed a reduction in values between 0.44 and 0.57 during
moderate geomagnetic storms. The lowest values of ApEn were noticed in the
following months: May 2010, March 2011 and January 2016. During major geomagnetic
storms as shown in Fig. 5, the minimum and maximum values of MLE are
respectively 0.03 and 0.04, implying a very strong reduction of chaotic
behaviour compared with minor and moderate geomagnetic storms. The lowest
values of MLE were found in the months of July 2012, June 2013 and March 2015.
Interestingly, further reduction in ApEn value (0.29–0.40) was as well
noticed during this period. Thus, during major geomagnetic storms, chaotic
behaviour and dynamical complexity subside significantly.

We display in Fig. 6 the results of MLE and ApEn computation for the
VB_{s} which has been categorized according to the periods of minor,
moderate and major geomagnetic storms. The values of MLE (bar plot) were
between 0.06 and 0.20 for VB_{s}. The result obtained indicates strong
chaotic behaviour with no significant difference in chaoticity during minor,
moderate and major geomagnetic storms. Similarly, the results obtained from
computation of ApEn (stem plot) for VB_{s} depict a minimum value of 0.60
and peak value of 0.87 as shown in Fig. 6. The ApEn values of VB_{s}
indicates high dynamical complexity response with no significant difference
during the periods of the three categories of geomagnetic storm
investigated.

The test for non-linearity in the *D*_{st} signals during minor, moderate and
major geomagnetic storms was analysed through the DVV analysis. Shown in
Fig. 7 is the DVV plot and DVV scatter plot during minor geomagnetic storms
for January 2009 and January 2014. We found that the DVV plots during minor
geomagnetic storms reveals a slight separation between the original and
surrogate data. Also, the DVV scatter plots show a slight deviation from
the bisector line between the original and surrogate data which implies
non-linearity. Also, during moderate geomagnetic storms, we notice that the
DVV plot depicts a wide separation between the original and the surrogate
data. Also, a large deviation from the bisector line between the original
and the surrogate data was also noticed in the DVV scatter plot as shown in
Fig. 8 thus indicating non-linearity. In Fig. 9, we display samples
of DVV plot and DVV scatter plot during major geomagnetic storm for October 2011
and December 2015. The original and the surrogate data showed a very large
separation in the DVV plot during major geomagnetic storm. The DVV
scatter plot depicts the greatest deviation from the bisector line between
the original and the surrogate data which is also an indication of
non-linearity. The DVV analysis of the VB_{s} time series during minor,
moderate and major geomagnetic storms shown in Figs. 10–12 revealed a
separation between the original and surrogate data with no significant
difference between the periods of minor, moderate and major geomagnetic
storms.

## 4.1 The chaotic and dynamical complexity response in *D*_{st} at minor, moderate and major geomagnetic storms

Our result shows that the values of MLE for *D*_{st} during minor
geomagnetic storm are higher, indicating significant chaotic response during
minor geomagnetic stormy periods (bar plot, Fig. 5). This increase in
chaotic behaviour for *D*_{st} signals during minor geomagnetic storms may
be as a result of asymmetry features in the longitudinal distribution of
solar source region for the corotating interaction regions (CIRs) signatures
responsible for the development of geomagnetic storms (Turner et al., 2006;
Kozyra et al., 2006). CIR-generated magnetic storms are generally weaker than
ICME- or MC-generated storms (Gonzalez et al., 1994; Tsurutani et al., 1995;
Feldstein et al., 2006; Richardson and Cane, 2011). Therefore, we suspect
that the increase in chaotic behaviour during minor geomagnetic storms is
strongly associated with the asymmetry features in the longitudinal
distribution of solar source region for the corotating interaction region
(CIR) signatures. For most of these periods of moderate geomagnetic storms,
the values of MLE decreases compared to minor geomagnetic storms. This
revealed that as geomagnetic stormy events build up, the level of
unpredictability and sensitive dependence on initial condition (chaos) begin
to decrease (Lorentz, 1963; Stogaz, 1994). The chaotic behaviour during
major geomagnetic storms decreases significantly compared with moderate
geomagnetic storms. The reduction in chaotic response during moderate and
its further declines at major geomagnetic storms may be attributed to the
disturbance in the interplanetary medium driven by sheath preceding an
interplanetary coronal mass ejection (ICME) or combination of the sheath and
an ICME magnetic cloud (Echer et al., 2008; Tsurutani et al., 2003; Meng et
al., 2019). Notably, the dynamics of the solar-wind–magnetospheric
interaction are dissipative and chaotic in nature (Pavlos, 2012), and the
electrodynamics of the magnetosphere due to the flux of interplanetary
electric fields had a significant impact on the state of the chaotic
signatures. For instance, the observation of strong chaotic behaviour during
minor geomagnetic storms suggests that the dynamics was characterized by a
weak magnetospheric disturbance. The reduction in chaotic behaviour at
moderate and major geomagnetic storm period reveals the dynamical features
with regards to when a strong magnetospheric disturbance begins to emerge.
Therefore, our observation of chaotic signatures at different categories of
geomagnetic storm has a potential capacity to give useful diagnostic
information about monitoring space weather events. It is important to note
that the features of *D*_{st} chaotic behaviour at different categories of
geomagnetic storm has not been reported in the literature. For example,
a previous study of Balasis et al. (2009, 2011) investigated dynamical
complexity behaviour using different entropy measures and revealed the
existence of low dynamical complexity in the magnetospheric dynamics and
attributed it to ongoing large magnetospheric disturbance (major geomagnetic
storm). The work of Balasis et al. (2009, 2011), where a certain dynamical
characteristic evolved in the *D*_{st} signal was revealed, was limited to
1 year of data (2001). It is worthy to note that the year 2001, according to
sunspot variations, is a period of high solar activity during solar cycle 23.
It is characterized by numerous and strong solar eruptions that were
followed by significant magnetic storm activities. This confirms that on
most of the days in year 2001, the geomagnetic activity is strongly
associated with major geomagnetic storms. The confirmation of low dynamical
complexity response in the *D*_{st} signal during major geomagnetic storms
agree with our current study. However, the idea of comparing the dynamical
complexity behaviour at different categories of geomagnetic storms and
revealing its chaotic features was not reported. This is the major reason why
our present investigation is crucial to the understanding of the level of
chaos and dynamical complexity involved during different categories of
geomagnetic storms. As an extension to the single-year investigation done by
Balasis et al. (2009, 2011) during a major geomagnetic storm, we further
investigated 9 years of data of *D*_{st} that covered minor, moderate and
major geomagnetic storms (see Fig. 5, stem plots) and unveiled their
dynamical complexity behaviour. During major geomagnetic stormy periods, we
found that the ApEn values decrease significantly, indicating reduction in
the dynamical complexity behaviour. This is in agreement with the low
dynamical complexity reported by Balasis et al. (2009, 2011) during a major
geomagnetic period. Finally, based on the method of DVV analysis, we found
that a test of non-linearity in the *D*_{st} time series during major
geomagnetic storms reveals the strongest non-linearity features.

## 4.2 The chaotic and dynamical complexity behaviour in the VB_{s} as input signals

The results of the MLE values for VB_{s} revealed a strong chaotic
behaviour during the three categories of geomagnetic storms. Comparing these
MLE values during minor to those observed during moderate and major
geomagnetic storms, the result obtained did not indicate any significant
difference in chaoticity (bar plots, Fig. 6). Also, the ApEn values of
VB_{s} during the periods associated with minor, moderate and major
geomagnetic storms revealed high dynamical complexity behaviour with no
significant difference between the three categories of geomagnetic storms
investigated. These observation of high chaotic and dynamical complexity
behaviour in the dynamics of VB_{s} may be due to interplanetary
discontinuities caused by the abrupt changes in the interplanetary magnetic
field direction and plasma parameters (Tsurutani et al., 2010). Also, the
indication of high chaotic and dynamical complexity behaviour in VB_{s}
signifies that the solar wind electric field is stochastic in nature. The
DVV analysis for VB_{s} revealed non-linearity features with no
significant difference between the minor, moderate and major geomagnetic
storms. It is worth mentioning that the dynamical complexity behaviour for
VB_{s} is different from what was observed for *D*_{st} time series data.
For instance, our results for *D*_{st} time series revealed that the
chaotic and dynamical complexity behaviour of the magnetospheric dynamics
are high during minor geomagnetic storms, reduce at moderate geomagnetic
storms and further decline during major geomagnetic storms. The
VB_{s} signal revealed a high chaotic and dynamical complexity behaviour
at all the categories of geomagnetic storm period. Therefore, these
dynamical features obtained in the VB_{s} as input signal and the *D*_{st}
as the output in describing the magnetosphere as a non-autonomous system
further support the finding of Donner et al. (2019) that found increased or
unchanged behaviour in dynamical complexity for VB_{s} and low dynamical
complexity behaviour during storms using the recurrence method. This suggests
that the magnetospheric dynamics are non-linear, and the solar wind dynamics are
consistently stochastic in nature.

This work has examined the magnetospheric chaos and dynamical complexity
behaviour in the disturbance storm time (*D*_{st}) and solar wind electric
field (VB_{s}) as input during different categories of geomagnetic storms.
The chaotic and dynamical complexity behaviour at minor, moderate and major
geomagnetic storms for solar wind electric field (VB_{s}) as input and
*D*_{st} as output of the magnetospheric system were analysed for the period
of 9 years using non-linear dynamics tools. Our analysis has shown a
noticeable trend of these non-linear parameters (MLE and ApEn) and the
categories of geomagnetic storm (minor, moderate and major). The MLE and
ApEn values of the *D*_{st} have indicated that the chaotic and dynamical
complexity behaviour are high during minor geomagnetic storms, low during
moderate geomagnetic storms and further reduced during major geomagnetic
storms. The values of MLE and ApEn obtained from VB_{s} indicate that
chaotic and dynamical complexity are high with no significant difference
during the periods of minor, moderate and major geomagnetic storms. Finally,
the test for non-linearity in the *D*_{st} time series during major
geomagnetic storms reveals the strongest non-linearity features. Based on
these findings, the dynamical features obtained in the VB_{s} as input and
*D*_{st} as output of the magnetospheric system suggest that the
magnetospheric dynamics are non-linear, and the solar wind dynamics are
consistently stochastic in nature.

The code is a collection of routines in MATLAB (MathWorks) and is available upon request to the corresponding author.

The data of disturbance storm time (*D*_{st}) are available at the World Data Centre for Geomagnetism, Kyoto, Japan: https://doi.org/10.17593/14515-74000 (World Data Center for Geomagnetism et al., 2015), while the solar wind electric field data (VB_{s}) are archived at the National Aeronautics and Space Administration (NASA), Space Physics Facility: https://omniweb.gsfc.nasa.gov/form/dx1.html (King and Papitashvili, 2004).

IAO analyzed the data; wrote the codes; and wrote the introduction, materials, and methods sections. OIO and OSB generated the idea behind the problem being solved and interpretation of results obtained, and OOO and ANN contributed to the discussion of the article.

The authors declare that they have no conflict of interest.

The authors would like to acknowledge the World Data Centre for
Geomagnetism, Kyoto, and the National Aeronautics and Space Administration (NASA)
Space Physics Facility for making the *D*_{st} data and solar wind plasma data available for research purposes. Also, the efforts of anonymous reviewers in ensuring that the quality of the work is enhanced are highly appreciated.

This paper was edited by Giovanni Lapenta and reviewed by two anonymous referees.

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*D*

_{st}indicate that chaotic and dynamical complexity responses are high during minor geomagnetic storms, reduce at moderate geomagnetic storms and decline further during major geomagnetic storms. However, the MLE and ApEn values obtained from solar wind electric field (VB

_{s}) indicate that chaotic and dynamical complexity responses are high with no significant difference between the periods that are associated with minor, moderate and major geomagnetic storms.

*D*

_{st}indicate that chaotic and dynamical complexity responses are...