1. What is linearity? What is nonlinearity?
Linearity refers to the proportion and linear relationship between quantity and quantity, representing regular smooth motion in space and time; Nonlinear refers to the non-proportional and nonlinear relationship, which is characterized by irregular movement and mutation. For example, how many times is the eyesight of two eyes more than that of one eye? It's easy to think it's twice, but it's actually 6- 10! This is nonlinearity: 1+ 1 is not equal to 2. The generation of laser is nonlinear! When the applied voltage is small, the laser is like an ordinary electric lamp, and the light scatters in all directions; When the applied voltage reaches a certain value, a new phenomenon will suddenly appear: the excited atoms seem to hear the command of "align to the right" and emit monochromatic light in the same phase and direction, that is, laser. The characteristic of nonlinearity is that it spans all professions and permeates all fields, which can be said to be "everywhere". For example, there is chaos in celestial motion; The oscillation of electricity, light and sound waves will suddenly fall into chaos; The geomagnetic field changed direction 16 times in 4 million years, also because of chaos. Even human beings are inherently nonlinear: contrary to traditional thinking, the EEG and heartbeat of healthy people are not regular, but chaotic, and chaos is the expression of vitality. Chaotic systems respond faster to external stimuli than non-chaotic systems. It can be seen that nonlinearity is around us and can't be hidden.
classify
Non-essential nonlinearity: nonlinearity that can be linearized by small deviation linearization method.
Inherent nonlinearity: nonlinearity that can not be solved by small deviation linearization method.
2. The meaning and difference between linearity and nonlinearity.
meaning
"Linearity" and "nonlinearity" are two mathematical terms. The so-called "linearity" refers to the proportional relationship between two quantities. If you draw it in a rectangular coordinate system, it is a straight line. A system described by a linear functional relationship is called a linear system. In a linear system, the sum of parts equals the whole. The equation describing a linear system follows the superposition principle, that is, the different solutions of the equation add up to the original solution. This is one of the most basic characteristics of linear systems. "Nonlinear" means that the relationship between two quantities is not a "straight line" relationship, but a curve in a rectangular coordinate system. The simplest nonlinear function is an unary quadratic equation, that is, a parabolic equation. Simply put, all nonlinear functional relationships, such as all polynomial functional relationships above the first power, are nonlinear. A system described by a nonlinear functional relationship is called a nonlinear system.
differentiate
Qualitative speaking, there is only one linear relationship, while the nonlinear relationship is ever-changing and endless. Linearity is a special case of nonlinearity, which is a simple proportional relationship and the contributions of each part are independent of each other; However, nonlinearity is a deviation from this simple relationship, and each part affects each other and has a coupling effect, which is the fundamental reason for the complexity and diversity of nonlinear problems. Because of this, the independence of various factors in the nonlinear system is lost: the whole is not equal to the sum of parts, the superposition principle is invalid, and the sum of two solutions of the nonlinear equation is no longer the solution of the original equation. Therefore, nonlinear problems can only be analyzed in detail.
The difference between linear and nonlinear phenomena generally has the following characteristics:
(1) In the form of motion, linear phenomena generally show smooth motion in time and space, which can be expressed by a functional relationship with good performance, while nonlinear phenomena show the transformation and jump from regular motion to irregular motion;
(2) The response of linear system to external influences is smooth and smooth, while the slight change of parameters in nonlinear system can cause the qualitative change of system motion form at some joints. A large number of interactions between nature and human society are nonlinear, and linear action is only an approximation of nonlinear action under certain conditions.