# Analysis of Electroencephalograms Based on the Phase Plane Method
## 2 Phase Plane Method  
A phase portrait is a graphical representation of a dynamical system, which is described by a system of differential equations. It allows us to visualize the change in the state of the system as a function of time. A phase portrait is constructed in a coordinate plane, where each point represents the state of the system at a certain point in time. The coordinate axes correspond to the variables of the system's state. The phase portrait allows us to analyze the behavior of the system, determine the stability of its states, and find points of equilibrium. It can also help in predicting the future behavior of the system and determine the optimal parameters to achieve the desired results. To build a phase portrait, it is necessary to:  
* Define the system state variables. These can be physical quantities such as position, speed, temperature, etc. ;
* Define differential equations that describe how the variables of the state of the system change over time. These equations can be derived from physical laws or experimental data;
* Solve the differential equations for different initial conditions to obtain the values of the variables of the state of the system at different points in time;
* Draw points corresponding to the values of the state variables on the coordinate plane. Each point represents the state of the system at a particular point in time;
* Connect the points with lines or curves to obtain a phase portrait. The shape and nature of these lines or curves can provide information about the behavior of the system. The phase portrait can be supplemented with additional elements such as phase trajectories, phase flows, and phase lines. These elements can help to visualize and analyze the behavior of the system in detail. A phase portrait is a graphical representation of the states of a system as a function of time. It consists of several basic elements: equilibrium points, phase trajectories, phase lines. The phase portrait is a powerful tool for analyzing and understanding the dynamics of a system. It is used in various fields of science and technology: mechanics and physics, biology, economics, and finance. In radio engineering, the phase portrait is used to analyze the dynamics of electrical circuits. For example, when analyzing the operation of a generator of electromagnetic oscillations, both harmonic and relaxation, the phase portrait allows us to determine different modes of operation. The phase plane method is a qualitative method for studying both the stationary and transient regimes, including nonlinear systems. In fact, this method is reduced to the numerical solution of a second-order differential equation. The main advantage of the phase plane method is its suitability for the analysis of nonlinear systems. Some important features of nonlinear systems, which are impossible or difficult to study analytically, can be studied using graphoanalytical constructions on the phase plane [5]. The phase trajectory is a set of points; the horizontal coordinate corresponds to the system function, and the vertical coordinate is its derivative at each moment of time. The phase portrait (PP) shows all possible system operation regimes at any initial conditions. In particular, it is possible to observe limit cycles, if they exist. Besides, the PP illustrates the stability of limit cycles, and the shape of limit cycles reflects the oscillations shape [6,7]. If the equations of a system are presented in canonical form, then the state vector of the system uniquely determines its state. Using the phase plane method, the behavior of the nonlinear system under study is considered and described, not in the time domain (that is, in the form of equations of processes in the system), but in the phase space of the system (that is, in the form of phase trajectories). Each state of the system in the state space corresponds to a point. The depicted point moves over time, describing the phase trajectory.