What Is Chaos Theory?
Chaos theory is a method with both qualitative thinking and quantitative analysis. It is used to discuss dynamic systems (such as population movement, chemical reactions, meteorological changes, social behavior, etc.) must be integrated and continuous instead of Only a single data relationship can explain and predict the behavior.
Chaos theory
(A method with both qualitative thinking and quantitative analysis)
- Another feature of chaos theory is its development characteristics. He has three principles:
- 1.Energy will always follow the path of least resistance
- 2. There is always a fundamental structure that is usually invisible. This structure determines the path of least resistance.
- 3. This fundamental structure, which always exists and is usually invisible, can not only be discovered, but also changed.
- Chaos theory is a method that combines qualitative thinking and quantitative analysis to explore dynamics
- American Meteorologist 1963
- The word chaos originally referred to the state of chaos before the universe was formed. Chinese and ancient Greek philosophers held the theory of chaos at the origin of the universe, claiming that the universe has gradually formed an orderly world from the beginning. In a well-ordered universe, western natural scientists, after a long period of discussion, have discovered many laws in nature one by one, as everyone knows well
- "Cotangent sequence" is a typical example of the butterfly effect. You see, each of the following three series is the cotangent of the previous one; the initial values are 1, 1.00001, and 1.001, but starting from the 10th item, the three series start to form huge differences. This is a chaotic sequence of numbers. After a sufficient number of items, the numbers obtained can be regarded as random and chaotic.
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- In theory, the function is continuous, and the real-world function is discontinuous. Therefore, the above problem (computer cannot always accurately calculate cot (a [n])) must be removed from a certain position. The COT function itself is discontinuous. In some places, there is a great difference in y due to a small gap in x.
- (1) Randomness: The system in chaos is an irregular behavior caused by the randomness of the dynamics inside the system, which is often called internal randomness. For example, in a one-dimensional non-linear mapping, even if the mathematical model describing the evolutionary behavior of the system does not contain any additional random terms, even if the control parameters and initial values are determined, the behavior of the system in the chaotic region still appears random. . This kind of randomness spontaneously arises inside the system, and has completely different sources and mechanisms than external randomness. Obviously, it is an inherent randomness and mechanism in the deterministic system. Local instability in the system is a characteristic of internal randomness and the reason for its sensitivity to initial values.
- (2) Sensitivity: The chaotic motion of the system, whether discrete or continuous, low-dimensional or high-dimensional, conservative or dissipative. Temporal evolution or spatial distribution has a basic characteristic, that is, the extreme sensitivity of the system's motion orbit to the initial value. This sensitivity, on the one hand, reflects the strong influence of random system motion trends in non-linear dynamic systems; on the other hand, it will also lead to the unpredictability of long-term time behavior of the system. The so-called "butterfly effect" proposed by meteorologist Lorentz is a prominent and vivid illustration of this sensitivity.
- (3) Fractal dimension: Chaos has fractal dimension, which means that the geometry of the system's motion orbit in phase space can be described by fractal dimension. For example, the fractal dimension of the Koch snowflake curve is 1.26; the Lorentz model describing atmospheric chaos is 2.06. The chaotic motion of the system is infinitely entangled, folded, and kinked in the phase space, forming a self-similar structure with infinite levels.
- (4) Universality: When the system tends to be chaotic, the characteristics displayed have universal significance. Its characteristics do not change due to differences in specific systems and differences in system motion equations. These systems are all related to Feigenbaum constants. This is an important universal constant = 4.669201609l0299097 ...
- (5) Law of scale: Chaos is an orderless state with periodicity. It has an infinite level of self-similar structure and a scale-free region. As long as the accuracy of the numerical calculation or the resolution of the experiment is sufficiently high, an orderly motion pattern of small-scale chaos can be found from it, so it has the property of scale law. For example, in the period doubling bifurcation process, the infinitely nested similar structure of the chaotic attractor, from the hierarchical relationship, has the self-similarity of the structure and the structural invariance under the scale transformation, thus showing orderliness.
- Chaos is not an accidental, individual event, but a variety of macro and micro systems that exist in the universe. Everything is chaos. Chaos is not an independent science. It promotes and depends on each other and other sciences, which leads to many interdisciplinary subjects, such as chaos meteorology, chaos economics, and chaos mathematics. Chaos not only has great research value, but also has practical application value, which can create wealth directly or indirectly. The theoretical purpose of studying chaos is multifaceted: reveal the nature of chaos (inherent randomness), characterize its basic characteristics, understand its dynamic behavior, and strive to control it so that it can be used by humans. In the past 20 years, chaos in engineering systems has gradually been changed from what is considered to be only a harmful phenomenon to a phenomenon that is considered to have practical application value. A large amount of research work in recent years has shown that chaos and engineering technology are more and more closely related, and it is widely used in biomedical engineering, kinetic engineering, chemical reaction engineering, electronic information engineering, computer engineering, applied mathematics, and experimental physics. Application prospects. In terms of application, it mainly includes synchronization of chaotic signals and confidential communication, chaos prediction, chaos neural network information processing, chaos and fractal image processing, chaos-based optimization methods, chaos bioengineering, weather systems, ecosystems, chaos economy, etc. In addition, the technology for controlling chaos has also been applied to the research work of neural networks, lasers, chemical reaction processes, fluid mechanics, nonlinear mechanical fault diagnosis systems, nonlinear circuits, astronomical mechanics, medical and physical systems with distributed parameters. At present, in some areas where chaos is very important and useful, the purposeful generation or enhancement of chaos has become a key research topic.
- There are already some examples of applications of chaos theory in educational administration, curriculum and teaching, educational research, and educational testing. Since the object of education is man, man is an individual who fluctuates at any time, and the process of education basically follows certain criteria and undergoes long-term interaction, so it is quite in line with the framework of chaos theory. Therefore, according to chaos theory, the education system is prone to produce unexpected results. This result may be positive or negative. Whether it is positive or negative, it is important that in addition to short-term observations, the effectiveness of education or education research should accumulate long-term data, analyze possible contexts from it, and increase the predictability of education effects, and Use it to expand the effectiveness of education.
- Chaos theory is an evolutionary theory in which a system suddenly changes from order to disorder.
- The discovery of chaos reveals a fundamental misunderstanding of the relationship between the law and the resulting behavior, that is, between the cause and the result. We used to think that simple causes must produce simple results (which means that complex results must have complex causes), but now we know that simple causes can produce complex results. We recognize that knowing these laws is not the same as predicting future behavior.
- This idea has been made very useful by a group of mathematicians and physicists, including William Ditto, Alan Garfinkel, and Jim Yorke The practical technique they call chaos control. In essence, the idea is to make the butterfly effect work for you. Small changes in initial conditions can produce a big change in subsequent behavior, which can be an advantage; all you have to do is make sure you get the big change you want. Correct
- For nearly half a century, scientists have discovered that even though many natural phenomena can be reduced to simple mathematical formulas, their behavior cannot be predicted. As meteorologist Edward Lorenz found, simple
- Chaos control was implemented by William Ditto, Alan Garfinkel, and Jim Yorke. This idea was turned into a practical technique, starting with small changes, and causing great changes in hope and thought.
- There are two kinds of chaos control methods. One is to use appropriate strategies, methods, and methods to effectively suppress chaotic behavior and reduce the Lyapunov exponent to eliminate chaos. The second is to select a track with desired behavior as the control target. In general, an infinite number of unstable periodic orbits in chaotic attractors are often used as the preferred target, the purpose of which is to convert the chaotic motion trajectory of the system to the desired periodic orbit. Different control strategies must follow the principle that the design of the control law must change the original system to a minimum, so as to minimize the impact on the original system. From this point of view, control methods can be divided into two categories: feedback control and non-feedback control. Feedback control is a very mature and widely used engineering design technology. It mainly uses the essential characteristics of chaotic systems, such as the sensitive dependence on the initial point, to stabilize the unstable orbits that already exist in the system. Generally speaking, the advantage of feedback control is that it does not need to use any information about a given controlled system except the system output or status, does not change the structure of the controlled system, and has good track tracking ability and stability. The disadvantage is that it requires a more accurate mathematical model and input objective function or orbit, which cannot be used directly when there is only observation data and no mathematical equation. Compared with feedback control, non-feedback control mainly uses a small external disturbance, such as a small drive signal, noise signal, constant offset or weak modulation of system parameters to control chaos. The design and use of this control method are very simple , But the stability of the control process cannot be guaranteed. In both cases, the stable solution of the system is obtained by slightly changing the chaotic dynamic system. In the realization of controlling chaos, making the most of the characteristics of chaos is very important for determining the control target and selecting the control method. The basic methods of chaos control are: OGY method, continuous feedback control method (external force feedback control method and delayed feedback control method), adaptive control method, and intelligent control method (neural network and fuzzy control), etc.