## 2 Materials and Methods
### Determinism and State Space Representation  
From a linear perspective, the erratic (or irregular) behavior of the one system response is due to a random external component. The chaos theory finds that a random input is not the only cause to get an irregular output. A simple deterministic equation, as with nonlinear chaotic systems, can generate irregular data without the contribution of any external input.  
In a deterministic system, according to the deployed dynamical systems theory [24], once having known its current state, the future states are entirely determined. The system dynamical and geometric properties all are to some extent included in the state space representation. Equations of motion that describe the behavior of a physical system as a function of time are deployed in the state space. Henceforth, the drawn trajectories in the state space represent the time evolution that the state vector is undergoing over time. For this reason, the state space or phase space can be used to get an approach to the rules that govern the dynamical system evolution.  
In an experimental setting, the system equations are not available. Usually, this is obtained solely with one system response projection in a one-dimensional space, a scalar time series, which represents the acquired measurements of the system in question. The state space reconstruction method aims to reconstruct the state vectors from the time series, so that the time evolution of these vectors replicates a dynamics equivalent to that of the original system, as shown in Figure 3. A time series reflects roughly the time evolution of a single dynamic variable, but as [PERSON] once said, the evolution of a variable depends on other variables of the system or, in other words, its value is within the history of other variables with which it interacts [25].  
Ideally, a first-order ordinary differential equation system taking action on the state space [2; 12; 26] defines a dynamical system. A set of possibly infinite states and certain transition rules that specify how the system moves from one state to another can describe several systems. For a finite-dimensional state space \(\mathbb{R}^{m}\), a state is defined by a vector \(\mathbf{x}\in\mathbb{R}^{m}\). Henceforth, the dynamics of the system can be described by an \(m\)-dimensional map or by an \(m\) first-order ordinary differential equation system called flow. In the first instance, such as the Henon map, the time is a discrete variable,  
Figure 3: State space reconstruction general overview.  
\[\mathbf{x}_{n+1}=\mathbf{F}(\mathbf{x}_{n}),\quad n\in\mathbb{Z}, \tag{1}\]  
while in the latter, as is the Lorenz flow, the time is a continuous variable,  
\[\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{x}(t)=\mathbf{f}(\mathbf{x}(t)),\quad t \in\mathbb{R}. \tag{2}\]  
For each initial condition \(\mathbf{x}_{0}\), or \(\mathbf{x}(0)\), the solutions to Equations (1) and (2), a sequence of points \(\mathbf{x}_{n}\), \(\mathbf{ox}(t)\), respectively, describe a dynamic system trajectory. With its time evolution, a typical trajectory can tend to infinity or be confined to a bounded region in the state space. All the initial conditions that lead to the same asymptotic behavior of the observed trajectory are known as the basin of attraction [27].  
The trajectory described in the state space, from a single observable, is a presumable indication of the presence of a deterministic behavior if the states are not arranged as a point cloud, despite an erratic appearance in the time domain. In a 2D phase diagram, the variable time, on the abscissa axis, is replaced by the observable value in a prior time determined by the parameter \(\tau\), as discussed in Section 1; in a 3D phase diagram, except for the first axis, all other axes pick up previous values of the observable variable separated from each other a time \(\tau\Delta t\).