What does nonlinear mean? What is the difference between linearity and nonlinearity?

Nonlinear is one of the typical properties of complexity in nature, so how much do you know about nonlinear? The following is what I have compiled about nonlinearity, I hope you like it!

What is nonlinearity?

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".

The difference between linear and nonlinear

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.

(1) linearity

The definition of linearity is generally carried out from two interrelated angles: first, the superposition principle holds: what if? l，？ 2 is two solutions of the equation, so what about A? l+b？ 2 is also a solution of it, in other words, the superposition of two states is still a 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.

② Nonlinear

After defining the meaning of linearity, the corresponding nonlinear concept is easy to define:

Its? ,? Define the nonlinear operator n (? ) for some a, b or? 、? Not satisfied with L(a? +b？ )=aL(？ )+bL(？ ) operator? That is to say, the superposition principle does not hold, that is? With what? There is a coupling between, to (a? +b？ ) operation, is it equal to the difference? And then what? Except for the operation, right? With what? The operation of cross term (coupling term), or? 、? It is discontinuous (with mutation or fracture) and nondifferentiable (with inflection point).

Second, as an equivalent? This kind of expression, we can understand nonlinearity from another angle: in description? In a certain set of physical variables of a system, which one is the system? The corresponding changes of this variable or other variables caused by the initial changes of a variable are out of proportion, in other words, the rate of change between variables is not constant, and the slope of a function does not exist or is unequal within its definition domain. Generally speaking, the first-order incremental relationship between physical variables is asymmetric in the domain of variables. It can be said that this symmetry breaking is the most basic embodiment of nonlinear relationship and the root of the complexity of nonlinear system.

These two expressions of nonlinear concept are actually equivalent. The failure of superposition principle will inevitably lead to the asymmetry of the relationship between its two physical variables; On the other hand, if the relationship between physical variables is asymmetric, then the superposition principle does not hold. The reason why the two expressions are used is that they have their own convenience for different objects in different occasions. For example, the former is convenient to study the relationship between the whole system and its parts and the properties of differential equations, while the latter will be convenient to study the relationship between specific variables (including the time behavior of variables).

Regarding the concept of nonlinearity, it should be emphasized that the formulation of linearity or nonlinearity is relative to physical variables, that is to say, only the relationship of physical variables is the basis for judging whether it is nonlinear, and the relationship of non-physical variables cannot be used as a criterion for nonlinearity. The physical variables mentioned here refer to those that are observable, interesting and meaningful to human beings. For example, in fractal theory, the fractal dimension d of a simple fractal is constant, and in the scale-free interval lnN=DlnL, the relationship between lnN and lnL is linear, but obviously it cannot be concluded that a simple fractal is linear. The physical variables here are n and l, not nN and lnL after logarithmic transformation, that is, n and l are observable, interesting and meaningful to people, not lnN and lnL. The relationship between n and l is nonlinear, so it can be concluded that fractal is nonlinear. For another example, the direct relationship between price and time (smaller than the scale-free nature of the cotton price index calculated by Mandbrolt) is exactly interesting and meaningful, and the relationship between them is nonlinear, so the change of price with time is a nonlinear phenomenon.

Properties of nonlinearity

Nonlinear science is in the process of development. The nonlinear universal classes it studied in specific scientific have been formed (such as chaos, fractal and soliton), are being formed (such as adaptability and spontaneous behavior) and will be formed, so the nonlinear properties have not been fully presented, and it is impossible to fully discuss the nonlinear properties here. Only from below? The relationship between nonlinearity and linearity? 、? Physical mechanism of nonlinearity? And then what? Nonlinear and stability? Three aspects are preliminarily discussed.

The relationship between nonlinearity and linearity

Nonlinear and linear are relative and contradictory concepts. On the one hand, they can be transformed into each other to a certain extent, on the other hand, they are essentially different, and they exist at the same time. In this system, the nature of the corresponding aspects of the system is defined.

① the close relationship between nonlinearity and linearity

First of all, some linear equations can be transformed into nonlinear equations in mathematics to solve. Some nonlinear problems in physics can also be studied by mathematical transformation into linear equations. For example, the nonlinear KdV equation is transformed into a linear integrable equation by scattering inversion method, so as to obtain an accurate analytical solution; Some weakly nonlinear problems can be transformed into some linear problems by linear approximation, which has been widely used in various disciplines and is quite effective.

Secondly, in some cases, the analytical solution obtained from the equation can not provide more information and help to better understand the behavior of the system, but from the nonlinear form of the solution, we can easily get the important properties of the system under study. For example, consider a simple equation: d2X/dt2+X=0, and its solution is X=Acos(t)+Bsin(t). From this nonlinear form, we can easily know that it is a periodic function and satisfies cos(t+2? )=cos(t)，sin(t+2？ ) =sin(t). However, from the analytical form of cos(t) and SIN (t), it is extremely difficult to prove that they are periodic. Therefore, it is considered that linear equations can get analytical solutions, while nonlinear equations are difficult to get analytical solutions, so it is inaccurate that linear performance gives more useful information than nonlinear. This means that it is not only possible but also necessary to study some problems from a nonlinear perspective.

So linearity and nonlinearity can be transformed into each other to some extent, which shows that there is a close relationship between linearity and nonlinearity.

② The essential difference between nonlinearity and linearity.

Although nonlinearity and linearity can be transformed into each other through mathematical transformation, they are related to each other in mathematics, but under the same perspective, level and frame of reference, nonlinearity and linearity are essentially different.

Mathematically, linear functional relationship is a straight line, and nonlinear functional relationship is a non-straight line, including various curves, broken lines, discontinuous lines and so on. The linear equation satisfies the superposition principle, but the nonlinear equation does not. Linear equations are easy to get analytical solutions, while nonlinear equations are generally not.

In physics, an approximate linear problem (not what we call a nonlinear problem) can be solved with a certain accuracy by linear approximation method, that is, according to the accuracy requirements of specific problems, several linear problems are solved in turn and superimposed, and a good approximate solution can be obtained. However, for nonlinear problems, the linear approximation method will fail because of the slow divergence and convergence of small parameters, especially for high-speed motion, strong interaction, long-term dynamic behavior and other nonlinear situations, the linear approximation method will be completely powerless. These limitations of linear approximation methods lead to the irreplaceability of nonlinear methods, which can only be used in places with strong nonlinearity that linear methods cannot handle. Linear approximation is not always effective, which is not only a methodological problem, but also a natural view problem. There are both quantitative changes and qualitative changes in nature, and in qualitative changes, nature has to go through jumps or turns, which linearity cannot tolerate.

③ The function of nonlinearity and linearity in the same system

There are both connections and essential differences between nonlinearity and linearity. They often exist in a system at the same time, which defines the properties of different aspects of the system. A system usually has both linear and nonlinear characteristics. First, in a given nonlinear system, its nonlinear nature determines whether and where its equilibrium structure or stability mechanism exists. Secondly, the linear nature of the system determines the small vibration law of the system about its equilibrium point (stable structure), that is, the linear expansion property of the system near the stable point.