Nonlinear Dynamical Systems
Nonlinear dynamical systems can be linearized, if the relationship between elements of this is information.
To formalize the interaction between elements of the system it is important to know the nature of the phenomenon. Identify the main parameter of the process. Then the nonlinear of the system can be described as a characteristic of the environment. In every moment of life the environment system is described by constants. Nonlinear constants enable us to obtain linear equations system.
Linearization algorithm for dynamic system
1.Сonstruction structural model of the dynamic system.
2. The study of the nature of the processes taking place
3. The verbal description of the characteristics of the system
4. Creating a mathematical model of the popular tool is the system. Identifying the causes of non-linearity of the system (usually the cause is a positive feedback)
4. Description of conditions in which the system functioning
5. Creating a mathematical model, are popular tools of the system of linear differential equations
6. Identifying the causes of non-linearity of the system (typically, the impact of the environment)
7. Linearization is available if you imagine the behavior of a dynamical system in discrete time (the image - a stroboscopic effect)
8. Characteristics of the system change over time, changes can be studied quantitatively by means of differentiation,
9. In a separate, fixed point in time, we may assume that the system froze. This can be interpreted as the absence of nonlinearity (the image - stopped reel of film in one frame)
10. Covering all moments of life of the system - combine parts into a whole.
11.linear differential equations
Xi=parameters describing the behavior of the system
Bi=free coefficients are independent of time and describe the environment (nonlineare)