radiation, resulting in the formation of charged particles [1]. The presence of free electrons within the ionosphere makes the region an important one for radio signal transmission and reception. The ionosphere allows for the propagation of radio waves beyond the theoretical line of sight by refracting signals over longer distances across the world. Satellite-to-earth communication signals have to pass through the ionosphere, where they experience delay. This delay is the largest source of error and uncertainty for single-frequency communication systems [2]. The ionosphere interacts with the neutral atmosphere through internal and external processes, resulting in complex nonlinear processes [3]. The total electron content (TEC) is the main ionospheric parameter that has an over-bearing influence on communication and navigation systems. The TEC, which is an important by-product of all forms of space-dependent GNSS data, is the number of electrons in an imaginary tube with a cross-sectional unit area that extends from a GNSS satellite to a GNSS receiver. The TEC can be used to monitor the ionosphere and provide an overall description of it [4]. The TEC is measured in units of \(10^{16}\) electrons per square meter in area, where \(10^{16}\) electrons/m \({}^{2}\) = 1 TEC unit [5]. The value of the TEC along the signal path is  
\[TEC=\int_{Recitur}^{Satellite}N_{e}\,ds \tag{1}\]  
where \(N_{e}\) is the electron density along the signal path. Doppler frequency shift (DFS) measurements have been an age-long technique for monitoring the ionosphere. The DFS is estimated from the energy spectra of received signals. Ref. [6] reported that the DFS quasi-periodic variations observed in their work are due to the passage of traveling ionospheric disturbances associated with acoustic-gravity waves (AGW) at the height of the F-layer of the ionosphere, and they concluded that DFS measurements can be an effective diagnostic tool for the analysis of ionospheric disturbances. According to [PERSON] [7], Doppler frequency shift measurements are of considerable interest because they reveal short-term variations in the ionization of the ionosphere rather than time-averaged values. These short-term variations have critical implications for the propagation of HF radio waves through the medium. Several researchers have investigated the response of ionospheric DFS to selected space weather events, such as geomagnetic activities, solar flares, irregularities, and meteorological events. For example, [8; 9; 10; 11; 12; 13]). [PERSON] et al. [14] examined ionospheric disturbances during x-class solar flares (2021--2022) using GNSS data and wavelet analysis. Their study investigated a novel approach for detecting solar influences on the ionosphere. Similarly, [PERSON] et al. [15] studied solar flare effects observed over Mexico during 30-31 March 2022. Their investigation highlighted very interesting results from the solar flare effect at lower and higher ionospheric heights. DFS observations are particularly useful for detecting and monitoring traveling ionospheric disturbances and short-period gravity waves (GWs) in the ionosphere [16]. Of interest is the possibility of utilizing Doppler shift measurements to investigate the complex interactions between the vertical total electron content (VTEC) and DFS. Studying the ionosphere requires an approach that is robust to the complex nature of the region. Nonlinear dynamics and chaos theory tools are particularly well-suited for the analysis of complexity in the ionosphere. Chaos is defined as the aperiodic long-termed behavior in a deterministic system that exhibits sensitive dependence on initial conditions [17]. Considering how the state of the ionosphere at the current time step does not determine its condition in another time step, it is intuitive to consider nonlinear approaches in the study of the ionosphere. First, most of these techniques do not consider the probability distribution of the time series. Second, these tools are robust to outliers, which are frequent in the ionosphere.