Introduction to chaos

Source: http://blog.csdn.net/kongdong/

Author: fasiondog

Reference: edited by Wang Dongsheng and Cao Lei, chaos, fractal and its application, published by China University of Science and Technology, 1995

Chaos is an important concept of modern science and a very important part of Non-Linear science. At the end of the 19th century and the beginning of the 20th century, the research by Pang galai and Li yagmonov laid the scientific foundation of chaos and inspired people to further explore related problems. As an emerging science, chaos is a breakthrough in understanding the irregularities in nature. It has penetrated into all sciences, and its influence on all sciences (including natural sciences, social sciences, and even philosophy) is equivalent to the influence of calculus on mathematical engineering science in the 19th century. People think that there are only three things that science will always remember in the 20th Century: Relativity, quantum mechanics, and chaos ".The theory of relativity eliminates the illusion of absolute space and time; quantum mechanics eliminates the Norton dream of a controllable measurement process; and chaos eliminates Laplace's fantasies about deterministic predictability.

Introduction to chaos

"Chaos" is a very striking research hotspot in modern times, setting off the third revolution in basic science since the theory of relativity and quantum mechanics. The concept of chaos in science is different from the understanding in classical philosophy and everyday language. Simply put, chaos is a kind of rule-free movement that determines the appearance of a system. Chaos theory is a non-linear dynamic chaos. It aims to reveal the simple laws that may be hidden behind seemingly random phenomena, in order to find the common law that is commonly followed by a large category of complex problems.

In 1963, Lorenz published the "decisive non-cyclical flow" article in the atmospheric science magazine, pointing out that there must be a connection between the inability to accurately react the climate and the inability of long-term weather forecasts, this is the relationship between non-cyclical and unpredictable. He also discovered chaos "extreme sensitivity to initial conditions ". This vividly uses the butterfly effect as a metaphor: when making a weather forecast, as long as a butterfly fan its wings, this disturbance, it will cause a very big difference in a long distance, which will make a long prediction impossible.

Based on the study in 1960s, the study of chaos began to enter the climax. In January 1971, scientists formally introduced the concept of a singular scalar (for example, Henon [see figure (1-1)] and Lorenz [see figure (2-2)] In the dissipation system. In 1975, J. York and T.Y lie proposed the scientific concept of chaos. Throughout the 1970s s, people not only did a deeper research on chaos in theory, but also worked hard to find the singular inspector in the laboratory. J. in his famous paper "period 3 means chaos", York points out that in any one-dimensional system, as long as there is a period 3, the system can also have other periods of length, it can also present complete chaos.

The discovery of chaos in a deterministic system has changed the perception that the universe is a predictable system. Using the deterministic equation, we can't find a stable pattern, but get a random result, which completely breaks the fantasy of Laplace deterministic predictability. However, we also found that many signals that were previously considered as noise are actually generated by some simple rules. These "noises" that contain internal rules are different from real noises, and their rules can be applied completely.

Figure (1-1) Henon

There is no definite definition of what chaos is, but with the deepening of research, a series of characteristics and essence of chaos are revealed, exact definitions of chaos that are complete and meaningful will be generated. At present, chaos is regarded as a non-cyclical order. It includes the following features:

 

1. chaos has inherent certainty. Although it looks like noise, unlike noise, the system is described by a completely definite equation without any random factor, however, the system will still show similar randomness;

2. Chaos has a fractal property. The aforementioned lorenz and Henon Uris all have a fractal structure;

3. Chaos is a non-cyclical order with scale immutability. The Feigenbaum constant system is also followed in the process of chaos caused by branching.

4. Chaos is also sensitive to initial conditions. As long as the initial conditions are slightly deviated or slightly disturbed, the final state of the system will be significantly different. Therefore, the long-term evolution of chaotic systems is unpredictable.

Significance of detecting chaos in the real system

Traditionally, signals are divided into two categories:

L The waveform of the deterministic signal at all times is fixed;

L The waveform of the random process is determined by the probability distribution.

Figure (1-2) Logistic ing and random noise

However, such classification ignores another extremely important signal-chaotic signal. The waveform of a chaotic signal is very irregular. It looks like noise on the surface, but it is actually produced by deterministic rules, which are sometimes very simple. It is such a simple rule that generates complex waveforms that inspires great interest. In figure (1-2), we show the chaotic and white noise signals generated by Logistic ing. On the surface, we cannot determine the noise and chaos. Exciting: Practice has proved that chaotic signals exist in a large number of physical systems and natural systems! Although the emergence of chaos makes it impossible for us to predict the long-term behavior of the system, we can use the law of chaos to predict the short-term behavior of the system, this is more effective than the traditional statistical method.

The Application of chaos in engineering can be divided into two categories:

(1) chaotic signal synthesis: generates chaotic signals similar to noise

(2) Chaotic Signal Analysis: checks whether a chaotic signal exists from a certain phenomenon.

In this article, we will mainly discuss the second topic. The existence of chaotic phenomena has been detected, which is very advantageous for us to have a deeper understanding of the characteristics of the system. In most cases, when we confirm that there is chaos in the system, we can use the principle of chaos to filter chaotic signals from useful signals to improve the signal-to-noise ratio, however, traditional filtering methods may sometimes be ineffective.

Several methods for detecting chaos

Natural systems (physical systems, chemical systems, or biological systems) can present chaos, which has been widely recognized, it also caused many scholars to try chaotic recognition in the laboratory or in natural conditions. However, in the experimental system, noise interacts with the dynamic characteristics dominated by the internal equation that determines the evolution of the system. Therefore, the experimental system will certainly have random input, this makes it difficult to identify chaos. Next we will briefly introduce several different methods to identify chaos.

 

Power Spectrum

Power spectrum is the statistic that most people are familiar with and most widely used to characterize the characteristics of complex time series. It splits complex time series into the superposition of sine vibrations of different frequencies. The power spectrum at a given frequency is proportional to the square of the sine wave coefficient of the frequency. A typical power spectrum consists of one or more peaks, which correspond to the main frequency of occurrence in the signal. In addition to these main peaks, other frequencies may also appear, but the amplitude is relatively low, and the power spectrum is usually distributed in a broadband band.

The wide-band power spectrum (most of which have overlapping peaks) is often associated with chaotic dynamics. However, unfortunately, the "noise" is also closely related to the broadband spectrum, so the appearance of the broadband spectrum is not enough to confirm the chaos relative to the noise.

 

Phase space reconstruction

The generation of chaos is the result of the combination of the overall stability of the system and the local instability. The local instability makes it sensitive to the initial value, and the overall stability makes it in the phase space (also known as the state space) it shows a certain fragtal structure, which is called a chaotic scalar. It is precisely this kind of precise structure that allows us to use it to distinguish noise from chaos, because the real noise still shows a mess in the phase space. Phase space reconstruction is a simple and practical technology, but it still has great limitations. This is because the phase space technology is used to observe the structure of the source machine. It relies on human eyes to identify the source machine. When the dimension of the source machine is higher than three dimensions, we will be helpless. In addition, not all chaotic phenomena have chaotic Uris (such as Logist ing ).

 

Li yazuov index and dimension

Some quantitative measurements of complex dynamic behaviors have been proposed for the study of nonlinear dynamics. The two most commonly used quantities are the Lyapunov Exponent and Dimension, which respectively measure the regularity and geometric structure of the dynamic behavior. The Li yafmonov index describes the ratio of system trajectory convergence or divergence. When a system has both positive and negative Li yafmonov indexes, chaos exists. In fact, one of the important functions of the Li yafmonov index is to judge the chaotic behavior of the system. The dimension here refers to the number of fractional dimensions of the chaotic machine. In the phase space, the number of dimensions reflects the number of variables required for motion in the phase space, while in the scalar, the number of dimensions indicates the amount of information required to portray the scalar.