## 3 Algebraic Complexity of the Schumann Resonance Amplitude Data  
### Calibration of the H-Rank Algorithm for a Synthetic Chaotic Time Series  
As mentioned previously, the H-rank algorithm can be used successfully not only to detect the order of a linear recurrence, but also to evaluate the algebraic complexity of any time series [24; 26]. The main purpose of this article is the analysis of Schumann resonance amplitudes measured at different geographical locations and the mathematical study of the correlations between different magnetometers. As shown in example 1, a proper selection of \(\varepsilon\) is crucial for a proper application of the H-rank algorithm. Therefore, before starting with the magnetic field data, we first demonstrate the applicability of the H-rank algorithm for a synthetic chaotic time series. Furthermore, we employ the model of the master-slave coupled (MSC) logistic maps [27] to show how a proper calibration of the parameters of the H-rank algorithm can help detect the algebraic complexity of the MSC model. In other words, we consider a system where the degree of synchronization between two subsystems (the master and the slave) can be controlled and show that the H-rank algorithm is able to efficiently determine that degree of synchronization. Synchronization of chaotic systems is a vibrant area of research in non-linear analysis [28]. The first observations of synchronization date back to 1665, when [PERSON] observed anti-phase coupling in two identical pendulum clocks [29]. It took several centuries before synchronization of chaos became an active area of research in theory [30], as well as in a variety of applications such as secure communication systems [31; 32; 33], coupled lasers [34; 35], neural networks [36], chemical oscillators [37], and a grid of power generators [38]. Different types of chaos synchronization can be observed in coupled chaotic systems: identical (also known as complete) synchronization [39], generalized synchronization [40], phase synchronization [41], amplitude envelope synchronization [42], lag synchronization [43], and anticipated synchronization [44]. Typically, generalized synchronization is observed for unidirectional coupling, when the first (driving) system forces or pulls the second (driven) one, but there is no back action. Such a situation is often called MSC coupling. The onset of generalized synchronization can be interpreted as the suppression of the dynamics of the driven system by the driving one, so that the'slave' passively follows its'master'. The governing equations for the MSC logistic maps are shown in Equation (4) [27]:  
\[ \begin{split}& X_{k+1}=r_{X}\cdot X_{k}\cdot(1-X_{k}),\\ & Y_{k+1}=r_{Y}\cdot q_{k}\cdot(1-q_{k}),\\ & q_{k}=\Delta\cdot X_{k}+(1-\Delta)\cdot Y_{k};k=0,1,2\ldots \end{split}  \tag{4}\]  
where \(x\) and \(y\) represent the master and the slave accordingly, and the parameter \(\Delta\) defines the magnitude of the MSC coupling. We choose the parameter of the logistic map \(r_{x}=3.9\) for the master and \(r_{y}=3.89\) for the slave, ensuring that both logistic maps operate in the chaotic mode [45]. The initial conditions \(x_{0}\) and \(y_{0}\) are set accordingly to \(0.1\) and \(0.2\). Thus, the two logistic maps generate completely different time series when \(\Delta=0\) (indicating that the two logistic maps are then uncoupled). The increasing value of \(\Delta\) results in MSC synchronization between the two logistic maps. The time series \((x_{k})\) and \((y_{k})\) are completely different at \(\Delta=0.1\) (Figure 1a). The \(150\) singular values of the Hankel matrix constructed from the time series \((x_{k}-y_{k})\) are shown in Figure 1b (with the parameter \(d\) set to \(150\)). It can be seen that the value of the highest singular value is \((\sigma_{k}^{2})=14.6058\) (Figure 1b). MSC signification between the two logistic maps at higher \(\Delta\) values is represented in Figure S2. A further increase in \(\Delta\) results in an almost identical synchronization of the MSC logistic maps (Figure 2).