## 3 Results and Discussion
### Complex Measures  
We conducted a nonlinear analysis on both VTEC data and DFS to assess the complexity and predictability of the systems. We also computed the time delay and embedding dimensions to evaluate the complexity measures. The statistics (time delay, embedding dimension, Lyapunov exponents, and correlation dimension) are shown in Table 1. We obtained a time delay value of 17 for VTEC measurements at both Lagos and Abuja. We observed a time delay of 8 for the DFS in 2020 and 2020-2021 and a time delay of 10 in 2021. The time delays observed for other locations include \(\tau=40\) for a mid-latitude location during 2001 and 2008 [19] and \(\tau=2\) for four months in 2008 at a high-latitude station [47]. Figure 7: Contour plots of the hourly values of VTEC for the year 2021 (**upper** and **lower** panels show the variation over Abuja and Lagos). The value obtained for VTEC in this study is close to the values of \(\tau=16\) reported over the 120\({}^{\circ}\) E meridian between 1996 and 2004 [20] and \(\tau\geq 18\) reported by [21] at Indian stations of Agatti (geomagnetic latitude 2.38 N), Mumbai (geomagnetic latitude 10.09 N), and Jodhpur (geomagnetic latitude 18.3 N). The embedding dimension \(m\) was also evaluated for the VTEC signals and DFS using the method of false nearest neighbors (Table 1). It was observed that the VTEC values at the two locations had identical embedding dimensions during 2020 and throughout the 2020-2021 period. However, in 2021, Lagos and Abuja obtained values of 18 and 6, respectively. This suggests a significant dynamical variation in 2021 at both locations. The values of embedding dimensions obtained in this study were found to be higher than those reported in other regions of the world [20; 21; 47]. Variations in location, data considered, dynamics of the study areas, and study methods may account for these differences. The embedding dimensions for the DFS varied between 10 and 11 for the three time periods under consideration. A positive largest Lyapunov exponent is indicative of chaos in the system. The inverse of the largest Lyapunov exponent is typically used as the system's predictive horizon. In Figure 8, the linear fit to Equation (7) is shown for both VTEC and DFS. It was observed that the linear fits for the VTEC signals showed positive slopes, which is not obvious in the DFS. The statistics for the fit, which represents the largest Lyapunov exponent, are presented in Table 1. The largest Lyapunov exponent values of 0.031, 0.021, and 0.035 were observed for VTEC in Lagos during the 2020, 2021, and 2020-2021 periods, respectively, while values of 0.011, 0.033, and 0.041 were reported for VTEC in Abuja during the same time period. These values are lower than reported values of 0.3369 [20], 0.22 [23], and 0.1 [22] but higher than 0.0045 [21]. However, they were found to be in the range of 0.023 [47]. A direct consequence of the largest Lyapunov exponent is the prediction horizon for the system, which is the inverse of the Lyapunov exponent. From the results obtained, the prediction horizon for VTEC in Lagos was 32, 48, and 29 h in the periods of 2020, 2021, and 2020-2021, respectively. The prediction horizon in Abuja was estimated to be 90, 30, and 24 h during the periods of 2020, 2021, and 2020-2021, respectively. The Doppler shift frequency in an HF radio signal (DFS) was found to have a negative exponent in 2020 and 2020-2021, while a low positive value (\(\approx\)0) was obtained for 2021. This suggests that the dynamics of the DFS are not chaotic. It is opened that DFS is a stochastic system. The state of the ionosphere is important, not only for understanding our solar system, but also for various aspects of our existence, including communications and weather systems. A short prediction horizon implies that it is impossible to make long-term predictions about the ionosphere; hence, it reduces our ability to plan for its effect on our communication and weather systems. It also implies that the ionosphere is a dynamic system with rapidly evolving states.