Chaos is a common and extremely complex phenomenon in a deterministic nonlinear system. It communicates the connection between order and disorder, determination and randomness. It is a new leap in human understanding of the world and a new technology that transforms the world. . At present, the development and control of the unique properties of chaos to apply to some challenging engineering technical problems has presented an exciting prospect. Because the chaotic state is a kind of movement that is always limited to a limited area, the orbit is never repeated, and the behavior is complex, it has a unique process. In a sense, chaos should be a science about process, not a science about state, but a science about evolution, not a science about existence. Therefore, using a computer to simulate chaos is to promote chaos research. An important basic method indispensable for rapid development. Therefore, this paper designs a chaotic circuit and uses MATLAB software to simulate the circuit.
Chaos theory is an active frontier field that has only developed in recent decades. It is an important branch of nonlinear science. Together with quantum physics and relativity, it is called the three important scientific discoveries of the 20th century. It is an orderly determined disorder and thus resembles a random phenomenon.
In terms of science and engineering, its research is of great significance. Scientifically, because the use of electronic circuits is easy to realize various nonlinear dynamic systems, and the electronic measurement is more convenient than the measurement of other physical quantities, the oscilloscope can be used to directly obtain the graph of the measured data; the computer can process the data to calculate various types Non-linear dynamic parameters, so the study of electronic chaotic circuits occupies an important position in the chaos research of nonlinear dynamic systems. In engineering, through the analysis of chaotic circuit theory, we can achieve a comprehensive grasp and rational understanding of chaotic circuits, which will help promote the popular chaotic communication (chaotic modulation technology and chaotic secure communication) and signal encryption in the world today. , Object anti-counterfeiting, scientific experiment and other applied research.
Chaos is a phenomenon that is common in nature and human social systems but is not easy to study. It is only due to the development of non-linear science and the improvement of computers that the study of chaos has become possible, and a preliminary theory has been formed, and then began to explore its practical application value. Chaos is introduced into the encryption system due to its sensitive dependence on the initial value and broadband noise-like characteristics. Since the U.S. Naval Laboratory used electronic circuits to achieve chaotic synchronization for the first time in 1990, the use of chaos to achieve secret communication has become the most competitive chaotic application research field in recent years. Scientists from all over the world have participated in fierce competition, each stepping up research on new chaotic systems and developing secure communication technology.
However, the performance of chaotic circuits and whether they are suitable for their own engineering tasks require us to make a more in-depth theoretical analysis of the existing chaotic circuit systems. First of all, a clear answer must be given to the most fundamental question, which is whether the system is indeed chaotic. This requires us to prove the existence of chaos for a chaotic circuit; then, how random the system is, which is also before the application of chaotic circuits. A problem that must be clear is that we need to calculate the Lyapunov exponent to measure the randomness of the signal generated by the chaotic circuit; the third important performance index is the structural stability of the chaotic circuit, that is, the system changes the circuit parameters under the external interference, and the chaos is Whether it can be easily destroyed, it is necessary to study the bifurcation phenomenon of chaotic circuit parameters.
1.2 The development status of chaos
Chaos is a unique form of motion of nonlinear dynamic systems. It exists widely in nature, such as physics, chemistry, biology, geology, and various scientific fields such as technical sciences and social sciences. The phenomenon of chaos reveals that there is a bridge between certainty and randomness, which has far-reaching significance in scientific concepts. As a dynamic behavior produced by a deterministic system, it has extremely rich dynamic characteristics. Although Chaos has rapidly become one of the research hotspots at home and abroad in recent years, it has gone through a long process from being known, accepted, researched to engineering application.
The chaos control problem was first proposed by Ott, Grebogi and Yorker of the University of Maryland in 1990, and its control method was later called the OGY method. In the same year, Oitto et al. used this method for the first time to achieve stable control of the unstable periodic orbits of chaotic orbits in a mechanical system composed of hysteretic elastic strips, which opened the prelude to chaos control research. Over the past ten years, international research on theories, methods and experiments of chaos control has developed rapidly. In many fields, such as electronics, optics, chemistry, biology and medicine, applied research has recently achieved a number of impressive The remarkable achievements have fully demonstrated its huge application potential.
In nature and laboratories, there are many kinds of nonlinear systems and chaotic behaviors. Therefore, the corresponding chaotic control and its applications are also colorful. At present, people's research on chaos is still focused on three types: time chaos, space chaos, and time and space chaos. Although the current chaos control methods and their applications have developed rapidly, most of them are still focused on the control and application of time chaos.
1.3 The significance of studying chaos
With the increasing popularity of computers and various communication networks, secure communication has become a research hotspot in related disciplines such as computer communication, networking, and microelectronics. In the late 1970s, the modern confidential communication technology based on the science of confidentiality entered the practical stage, which was marked by the formulation and wide application of the DEST (DataEncryPtion Standard) standard.
In the information age, the requirements for information confidentiality are getting higher and higher. Intemet, as a commonly used information transmission medium, although it has a general confidentiality function, its confidentiality is far from meeting the needs of modern communication. In Internet communication, the phenomenon of information loss of confidentiality occurs from time to time, and the economic loss caused by this is difficult to estimate. Modern confidential communication is a communication method developed with military, diplomatic and commercial needs. It has been in the field of modern communication for more than 100 years. In wired (metal cable) and wireless communication, quite a few The incidents of leaking, eavesdropping, and deciphering have caused significant military and economic losses to some countries, and even led to the defeat of the war and heavy casualties. The consequences are very serious. With the application of secure communication systems, the situation has improved, but illegal intruders of computers and networks have not been eliminated. In order to achieve the purpose of stealing important information, some people use high-speed computers and advanced technology to try to crack the existing confidential communication system. It can be seen that the research and development of a more secure and secure communication system will be a long-term and arduous task.
In 1987, Fujisaka and Yhmata's research on chaotic synchronization and Pecora and Carroll's experimental research on chaotic synchronization in 1990 attracted widespread attention. This breakthrough development made it possible for the application of chaos theory to communication, and created chaotic synchronization in A new stage of application in confidential communication. This is a dynamic method, because its processing speed has nothing to do with the key length, so the calculation efficiency of this method is very high. The information encrypted by this method is to modulate the information signal into a chaotic signal that is almost completely random. Only when the receiver is modulated to the same or a small range of the specified circuit parameters of the transmitter, the two are synchronized with each other, and the information can be It is restored, compared with the existing encryption method, chaotic secure communication has a high degree of security, especially it can be used for real-time signal processing, but also suitable for static encryption. Although the current research on this new technology is still in the laboratory stage, due to its obvious advantages such as strong real-time, high confidentiality, and fast computing speed, it has shown its strong vitality and application prospects in the field of secure communications. The direction with extremely high research value has become one of the current frontier research hotspots in the field of communications.
Chaos is an externally complex performance caused by introspective randomness in a nonlinear deterministic system, and it is a pseudo-random motion that looks random. One of its basic characteristics is the extreme sensitivity of the system to initial conditions, that is, small differences in initial conditions will increase exponentially over time, which is ultimately unacceptable. Its long-term behavior is obviously random, uncontrollable and unpredictable. People's research on chaos began in the 1970s. American scientist Lorenz, known as the "father of chaos", once gave a popular definition: a real physical system, after eliminating all random effects, still has seemingly random performance, then this system is chaotic. . This definition of Lorenz states the following basic characteristics of chaos: (l) Chaos is an inherent characteristic of the system. The complexity shown by the system is caused by the system itself and by internal factors, not caused by external interference, but is a manifestation of the inherent randomness of the system. (2) Chaos is deterministic. The certainty of chaos is divided into two aspects: first, the chaotic system is a definite system, a real physical system; second, the performance of chaos is seemingly random, not truly random, and the state of the system is affected at every moment. The influence of the previous state is determined to appear, rather than randomly appearing like a random system. The state of a chaotic system can be completely reproduced, which is different from a random system. (3) The performance of chaotic systems is complex. The performance of a chaotic system is seemingly random, it is not a periodic motion, nor is it a quasi-periodic motion, it has good autocorrelation and low-frequency broadband characteristics.
The term chaos was first proposed by Li Tianyan (LI TY) and York (Yorke JA) in 1975. In 1975, they gave the Li-Yorke theorem in the article "Period 3 means chaos", and formed a special definition of chaos from the theorem:
Li-Yorke Theorem:
Let f(x) be a continuous self-mapping on [a, b]. If f(x) has 3 period points, then for any positive integer n, f(x) has n period points.
The definition of Li-Yorke chaos: the continuous self-mapping f(x) on the closed interval I can be determined to have chaos if the following conditions are met:
(l) The period of the period point of f(x) has no upper bound:
(2) There is an uncountable subset S on the closed interval I, satisfying
(i) For any x and y belong to S, when x is not equal to y, there is
(Ii) For any x and y belong to S, there is
(Iii) For any x belonging to S and any periodic point y, there are
According to the above theorems and definitions, for the continuous function f(x) on the closed interval I, if there is a periodic point with a period of 3, there must be any positive integer periodic point, that is, chaos must appear. In Li Tianyan's words, as long as there is cycle 1, any cycle can be "messy".
This definition is proposed for a set, but it shows the important characteristics of chaotic motion: first, there are periodic orbits of all orders; second, there is an uncountable set, this set contains only chaotic orbits, and any two orbits Neither tending to be far away nor approaching, but two states alternately appear, and any orbit does not tend to any orbit, that is, there is no asymptotically periodic orbit in this set. Third, chaotic orbits are highly unstable.
In 1983, Professor Cai Shaotang first proposed the famous Chua’s circuit, which is by far one of the most effective and simple chaotic oscillator circuits that produce complex dynamics in nonlinear circuits. By changing the parameters of Chua’s circuit, it is possible to produce very rich chaotic phenomena ranging from period-doubling bifurcation, single scroll, period 3 to double scroll, so that people can easily understand the chaotic mechanism and characteristics from the perspective of the circuit. the study. Initiated the study of chaos in our country. In 1986, the first China Chaos Conference was held in Guilin, which promoted the extensive research on chaos science in my country. In the same year, the Chinese scholar Xu Jinghua was the first in the world to propose a composite network of three kinds of nerve cells and proved the existence of chaos in it.
In the 1990s, chaos theory and other sciences penetrated extensively, including philosophy, mathematics, physics, chemistry, electronic technology, information science, astronomy, meteorology, economics, and even music, art and other fields. It is particularly worth emphasizing that in my country’s National Climbing Plan from 1991 to 1995 on the "non-linear science" important project, chaos research was ranked fourth among the 30 projects, which fully demonstrates the scientific community’s understanding of chaos. The degree of attention. The above is a brief overview of chaos. For specific generation and control methods, please refer to the literature.
2.2 Characteristics of Chaos
From the perspective of phenomenon, chaotic motion looks like a random process, but in fact, chaotic motion is essentially different from random process. Chaotic motion is caused by the internal characteristic of deterministic physical laws, and is derived from the external manifestation of internal characteristics, so it is also called deterministic chaos; while random processes are caused by external noise. The following describes the characteristics of chaos in detail.
(l) Inherent randomness
The steady state of chaos is not the three states that determine the motion under the usual concept: stationary, periodic motion and quasi-periodic motion, but a complex form of motion that is always limited to a limited area and the orbit is never repeated. First, chaos is inherent, and the complexity of the system is the system itself, determined by internal factors, not generated under external interference, but a manifestation of the internal randomness of the system. Second, the randomness of chaos is deterministic. The determinism of chaos is divided into two aspects. First, the chaotic system is a definite system, a real physical system; secondly, chaos and its theoretical chaotic performance is seemingly random, rather than truly random. The state at all times is affected by the previous state, and it appears definitely, rather than appearing randomly like a random system. The state of a chaotic system can be completely reproduced, which is different from a random system. Third, the performance of chaotic systems is complicated. The performance of a chaotic system is seemingly random, it is not a periodic motion, nor is it a quasi-periodic motion, it has good autocorrelation and low-frequency broadband characteristics.
(2) Long-term unpredictability
Since the initial conditions are limited to a certain limited accuracy, and small differences in the initial conditions may have a huge impact on the future evolution of time, it is impossible to predict the dynamic characteristics beyond a certain point in the future for a long time. That is, the long-term evolution behavior of a chaotic system is unpredictable.
(3) Sensitive dependence on initial value
As time goes by, the initial conditions that are arbitrarily close will show their own independent time evolution, that is, sensitive dependence on the initial conditions.
(4) Universality
When the system tends to chaos, the displayed characteristics are universal, and the system does not change due to differences in specific systems and differences in system motion equations.
(5) Fractal
The word Fractal is a new term used by BB Mandelbrot to create fractal geometry in the 1970s. It refers to the behavioral characteristics of chaotic motion trajectories in phase space. The motion trajectory of the chaotic system in the phase space undergoes infinite folding in a certain finite area, which is different from the general deterministic motion and cannot be expressed in general geometric terms. The fractal dimension can just represent this infinite folding . Fractal dimension means that the chaotic state of motion has a multi-leaf and multi-layer structure, and the leaf layer is more and more finely divided, showing an infinite level of self-similar structure.
(6) Ergodicity
Ergodicity is also called confounding. The "steady state" of chaos is not the three steady states of deterministic motion under the usual concept: static (balance), periodic motion, and quasi-periodic motion, but a kind of always confined to a limited area, never repeating the orbit, and complex behavior movement. Therefore, as time goes by, the trajectory of chaotic motion never stays in a certain state and traverses every point in the regional space.
Through the previous theoretical analysis, we have basically understood the basic structure of the chaotic circuit, and we designed the following basic system block diagram as follows:
Pic 4-1
The following mainly introduces the function of each module and the configuration of parameters:
·Product
In the system, we used a multiplier, the input is 3 bits, and its main parameters are as follows:
Figure 4-2
The main function of this multiplier is to multiply three inputs.
·Gain
The main parameter settings are as follows:
Figure 4-3
Here the signal gain is set to 0.1 , that is, the signal amplitude is multiplied by 0.1 as the signal output. Here we have used multiple signal gain modules, mainly 0.1 gain, 0.25 gain, 5.5 gain and -7 gain.
·integrator
The integrator module, 1/s, mainly plays an integral role on the left and right, and uses the Laplace transform, that is, multiplies the Laplace domain and 1/s to achieve the integral effect.
The main parameter settings are as follows:
This module is relatively familiar, and is mainly used for the waveform display function, so I won't introduce it here.
·XY Graph
This is mainly used for the display of waveforms, and is similar in function to the plot function, so I won't introduce it here.
4.3 Chaos circuit simulation results
Through the analysis in the previous section, we have basically obtained the basic simulation model of the chaotic circuit. The following gives our simulation results on this model.
Click the XY Graph module, you can get the following simulation results. The figure below shows that we are looking for the horseshoe in the attractor. We use the knowledge of topological horseshoe theory to strictly prove that the attractor is indeed a chaotic attractor.
Figure 4-5
According to the characteristics of the chaotic attractor in the chaotic motion, the chaotic attractor is the product of the combination of overall stability and local instability, and its performance in the phase space is "elongation" and "folding". It has a complex structure of stretching, folding and stretching, which keeps the exponentially diverging system in a limited space, that is, all movements outside the attractor move closer to the attractor, corresponding to a stable direction; and everything arrives The motion orbits inside the attractor repel each other and face the unstable direction. That is to say, on the whole, the system is stable, that is, all motion outside the attractor will eventually converge to the attractor; but locally, the motion inside the attractor is unstable, that is, adjacent motions. The orbitals are mutually exclusive and separated exponentially.
From the experiment, it is easy to observe the phenomenon of double period and quadruple period. A little change will result in a single-vortex chaotic attractor, which is more obviously a three-period window. Observing these windows indicates that the chaotic solution is obtained, not noise. At the end of the adjustment, I saw that the attractor suddenly filled the space occupied by the original two chaotic attractors, forming a double-vortex chaotic attractor. Since every point on the oscilloscope corresponds to every state in the circuit, the appearance of a double chaotic attractor means that when the circuit is in this state, it is equivalent to the initial response state of the circuit, and which state it will eventually reach depends entirely on the initial state. condition.
In addition, observing SCOPE, we can get the time-domain diagram of the entire chaotic circuit.
Figure 4-3
Through the above simulation conclusions, we do a simple analysis on the simulation results. The system outputs a square wave, thus verifying the correctness of the system.