How to judge linearity and nonlinearity

The difference between nonlinearity and linearity is as follows

Nonlinear, that is, the mathematical relationship between variables, is not a straight line but a curve, surface or uncertain attribute, which is called nonlinear. Nonlinear is one of the typical characteristics of complexity in nature. Compared with linearity, nonlinearity is closer to the essence of objective things and is one of the important methods to quantitatively study and understand complex knowledge. Any relationship that can be described by nonlinearity is generally called nonlinear relationship.

In a narrow sense, nonlinearity refers to the quantitative relationship that is not proportional and linear, and cannot be expressed in linear form, such as curves and surfaces. The independent variable in a broad sense changes in a special form, which is different from the traditional mapping relationship, such as the function of iterative relationship. The mapping of the last calculus is the independent variable of the next calculus, which obviously cannot be described and described by the usual linear function. Obviously, the changing law of natural things is not like a simple function image, and there is not a one-to-one relationship between them. If the linear relationship is independent and irrelevant, then the nonlinear relationship reflects the interaction. It is this interaction that makes the whole no longer simply equal to the sum of parts, but may have gains and losses different from "linear superposition".

Compared with linearity, nonlinearity negates linearity, and linearity is a special case of nonlinearity. Therefore, in order to understand the concept of nonlinearity and clarify what nonlinearity is, we must first clarify what linearity is, and then we must explain the definition of nonlinearity from two aspects: mathematical expression and physical meaning, so as to understand the concept of nonlinearity more completely.

The definition of linearity is generally carried out from two interrelated angles: First, the superposition principle holds that: "If ψl and ψ2 are two solutions of the equation, then aψl+bψ2 is also one of its solutions, in other words, the superposition of two states is still one state." The superposition principle means that there is no nonlinear interaction between subsystems of the studied system. Secondly, the functional relationship between physical variables is a straight line, and the rate of change between variables is constant, which means that the slope of the function exists everywhere and is equal in its definition domain, and the proportional relationship between variables is symmetrical in the whole definition domain of variables.