## 4 Conclusions  
Chaos theory has evolved over the past few decades due to improved techniques, the robustness of its processes, and the application of the results. There have only been a few attempts to apply chaos theory to the study of the ionosphere. This study extends the work of previous authors by considering two tropical locations close to the equator over a  
Figure 12: Monthly variation in transfer entropy values between VTEC and DFS at **(a)** Lagos and **(b)** Abuja.  
Figure 13: Seasonal variation in transfer entropy between VTEC and DFS at **(top)** Lagos and **(bottom)** Abuja.  
period of two years in relation to the quality of the HF radio signals transmitted between the two study areas. Our results showed that the VTEC at the two locations had the same embedding dimension in both locations for all the periods under consideration. Using both Lyapunov exponent and correlation dimension measures, we conclude that the dynamics of the VTEC at the two locations are chaotic in nature. However, the signal quality between these two locations was found to be a stochastic process. The influence of the ionosphere on propagated HF radio signals between the two locations was investigated using transfer entropy and correlation analysis. A negative correlation was observed at the beginning of the year, which weakened during the middle of the year, predominantly during the June solstice, before returning to a negative correlation towards the end of the year. The use of correlation measures in this study is for comparison with other nonlinear measures while taking into consideration the strength of the correlation values. Our discussion highlights the weakness of the correlation values themselves. However, the transfer entropy analysis revealed a weak nonlinear relationship between the two variables in January. This increased to a peak in April during the 2021 and 2020-2021 periods. Transfer entropy provides direct evidence that the VTEC influences radio signals in the ionosphere. This causality was observed to change during different months of the year. This means that when there are changes in the ionosphere, there is a corresponding change in the signal that passes through it. Thus, the state of the ionosphere is important for radio communication. The observed linear and nonlinear relationships between the ionosphere and signal quality at the two locations can be incorporated into communication planning for high-frequency radio systems at both annual and monthly scales. This study can be extended to other regions, especially high-latitude stations. Furthermore, the relationships at smaller temporal scales can be explored. There are other nonlinear techniques that can reveal more features of the VTEC that can be considered in future studies.  
Conceptualization, A.A., B.R. and S.O.; investigation, A.A., B.R., S.O., D.O., A.K., B.N., O.B., I.F., E.O., B.O., J.A. and O.A.; resources, A.A., B.R., S.O., D.O., A.K., B.N., O.B., I.F., E.O., B.O., J.A. and O.A.; writing--original draft preparation, A.A., B.R. and S.O.; writing--review and editing, A.A., B.R., S.O., D.O., A.K., B.N., O.B., I.F., E.O., B.O., J.A. and O.A. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.