## 3 Results and Discussion
### Complex Measures
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Period** & \(m\) & \(\tau\) & \(\lambda\) & \(d_{2}\) \\ \hline VTEC (Lagos) & 2020 & 11 & 17 & 0.031 & 1.553 \\ VTEC (Lagos) & 2021 & 18 & 17 & 0.021 & 1.775 \\ VTEC (Lagos) & 2020–2021 & 18 & 17 & 0.035 & 1.593 \\ \hline VTEC (Abuja) & 2020 & 11 & 17 & 0.011 & 1.485 \\ VTEC (Abuja) & 2021 & 6 & 17 & 0.033 & 1.388 \\ VTEC (Abuja) & 2020–2021 & 18 & 17 & 0.041 & 1.610 \\ \hline DFS & 2020 & 10 & 8 & \(-\)0.010 & 0.530 \\ DFS & 2021 & 11 & 10 & 0.002 & 0.312 \\ DFS & 2020–2021 & 10 & 8 & \(-\)0.009 & 0.390 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistics of complex measures, including the embedding dimensions (\(m\)), time delay (\(\tau\)), largest Lyapunov exponent (\(\lambda\)), and correlation dimension (\(d_{2}\)), of NASA VTEC and DFS at Lagos and Abuja during different time periods. In Figure 9, the slope (\(D_{2}\)) of Equation (9) for the parameters is presented with the statistics shown in Table 1. It can be observed that all the parameters observed showed non-integer values of \(d_{2}\). Specifically, \(d_{2}\) was found to be in the range \(1.553\leq d_{2}\leq 1.775\) in Lagos. Since \(d_{2}\) gives an indication of the number of variables required to model a system, it follows that VTEC in Lagos can be captured as a two-dimensional system. It could be inferred that, due to the closeness of values, the dynamics observed over the 2020-2021 period are driven by the dynamics of the VTEC during 2020. This could also be observed in their respective Lyapunov exponent values. In Abuja, there is a range of \(1.388\leq d_{2}\leq 1.610\), which implies that the dynamics of the VTEC at this location can be described by two variables. The correlation dimension values obtained in this study were found to be lower than those reported in other studies. Specifically, the estimation of the correlation dimension for the VTEC in other locations across the world yielded values of 2.78 [19], 3.6 [20], 2.74 [21], and 2.8-3.5 [22]. The differences in results could be attributed to our location of choice, period of observation, and length of time for which data were considered. The dynamics of geospace have been of interest to scientists because of their importance in space infrastructure, satellite communication signals, and terrestrial weather. Several studies have demonstrated that complex structures, not linear regimes, govern activities in different layers of geospace. This is due to the mixing of gases, coupling between different layers, and several driving forces acting differently across the layers of geospace. It has been shown that chaotic dynamics exist in the magnetosphere [18], stratosphere [48],  
Figure 8: Linear fit to \(d_{i}(k)=\alpha\epsilon^{\lambda k}\) (Equation (7)) for VTEC (TECU) at (**a**) Lagos and (**b**) Abuja, as well as for (**c**) DFS (Hz). Figure 9: Scaling plots for correlation dimensions for VTEC at (**a**) Lagos and (**b**) Abuja, as well as for (**c**) DFS. mesosphere [49], and ionosphere [19]. The implication of chaotic dynamics in geospace is that long-term prediction of the system is difficult. The state of the ionosphere is very important due to its huge impact on earth's satellite radio communications and the attendant effects of satellites domiciled in the region, as well as their associated space-dependent services. The short prediction horizon implies that the predictability of the ionosphere is only guaranteed for a short term. It also implies that the ionosphere is a dynamic system with rapidly evolving states, which requires effective monitoring and periodical forecasting. The results obtained in this study are in agreement with those reported for other regions of the world. We have shown that the ionosphere is a chaotic system at two locations within Nigeria. Our results showed a prediction horizon between 24 and 90 h for the VTEC at the two locations, while at least two variables were needed to describe their dynamics.