In this study, we examine the magnetospheric chaos and dynamical complexity
response to the disturbance storm time (

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,

The

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

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 (

The

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.

The method of average mutual information (AMI) is one of the non-linear
techniques used to determine the optimal time delay (

In determining the optimal choice of embedding dimension (

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

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

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
(

In this study,

Samples of

Samples of solar wind electric fields (

The plot average mutual information against embedding time delay
(

The plot of percentage of false nearest neighbours against
embedding dimension (

The results of MLE (bar plot) and ApEn (stem plot) for

The MLE (bar plot) and ApEn (stem plot) of

The MLE (bar plot) and ApEn (stem plot) of solar wind electric
field (

We display in Fig. 6 the results of MLE and ApEn computation for the

The test for non-linearity in the

The DVV plot and scatter plot for

The DVV plot and scatter plot for

The DVV plot and scatter plot for

The DVV plot and scatter plot for

The DVV plot and scatter plot for

The DVV plot and scatter plot for

Our result shows that the values of MLE for

The results of the MLE values for

This work has examined the magnetospheric chaos and dynamical complexity
behaviour in the disturbance storm time (

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 (

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

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