1Introduction

After decades of development, people are no longer strange to chaos, but how to correctly understand and study chaos and its application is still the main task of academic circles. As we all know, chaos is nonlinear, while what we know well is linear. What is the relationship between them? The discussion begins here.

1.1Linear, Nonlinear, and Chaos

Linear and nonlinear are used to explain the relationship between function y = f(x) and an independent variable x. To be specific, linear refers to the proportional relationship between two independent variables. It represents regular and smooth movement in space and time. However, nonlinear refers to the disproportionate relationship between two dependent variables. It represents irregular and mutation movement in space and time. For example, how many times the visual acuity of two eyes is that of an eye? It is very easy to think that it is two times, but actually, it is 6–10 times. This is the nonlinear relationship. Calculation result of one plus one is not two.

Although nonlinear relationship is ever changing and complicated, there is something in common that is different from linear relationship. Linear relationship is unrelated independence, while nonlinear relationship is interaction. It is the interaction that makes the entirety, which is not simply equal to the sum of the parts. On the contrary, unlike linear superposition, gain or loss may occur. For example, the generation of laser is nonlinear. Laser scatters in all directions like lamp when the voltage is small. When the applied voltage reaches a certain value, a new phenomenon applied will appear suddenly. The excited atoms scatter a kind of homogeneous light whose emit phase and direction are consistent as if they received an order to look right. This kind of homogeneous light is laser. For another example, the frequency components of linear signal remain the same, but nonlinear relationship leads to changes of frequency configuration. As long as there is nonlinearity, even if it is a small nonlinearity, it will produce sum frequency, difference frequency, frequency doubling, and other components. Behavioral mutations are caused by nonlinear relationships, but small deviation for the linear relationship doesn’t lead to behavioral mutations in general. And according to the original linear conditions, the small deviation can be described and comprehended by modified linear theory. System behavior may change suddenly when nonlinear relationship reaches a certain degree. Mutations often occur at a series of parameter thresholds of the nonlinear system, and each mutation is accompanied by a new frequency component. The system will eventually enter the chaotic state.

According to the characteristics of the nonlinear relationship, it can be concluded that if there exists chaos, the system must be a nonlinear system. There is a mutation behavior when a nonlinear system turns into the state of chaos. How can we know whether a system has turned into the state of chaos? How to judge a system that presents phenomenon for a long period of time? How to distinguish whether a system is subject to external random disturbances? All those questions are important issues in chaos research.

Chaos theory belongs to nonlinear science. Nonlinear scientific research always seems to make people’s understanding of “normal” things and “normal” phenomenon to explore the “abnormal” things and “abnormal” phenomenon. For example, solitary wave is not a regular propagation with periodic oscillation. Unconventional edit methods will be taken when a large number of unconventional phenomena occur when using multimedia technology for information storage, compression, transmission, conversion, and control process encountered. Chaos broke the convention that the future movement of a deterministic equation is determined strictly by the initial conditions and led to a so-called strange attractor phenomenon.

In nonlinear science, chaos is not exactly the same as what it means. Chaos phenomenon refers to a determined but unpredictable motion in nonlinear science. The external manifestations of chaos are very similar to purely random motion. That is, they are all unpredictable. Comparing with random motion, chaos motion is determinate. And its unpredictability is derived from the instability of the system of internal movement. Chaos system is sensitive to initial conditions and little change, no matter how small they are. Therefore, after a period of time evolution, the system will deviate from its original direction of evolution. Chaos is a common phenomenon in nature, such as the change of weather. An important feature of chaos is butterfly effect. So chaos means a nonperiodic motion generated by a deterministic system. Chaos, fractals, and soliton are the three most important nonlinear science concepts. Chaos theory is a part of nonlinear science, and only nonlinear system can produce chaotic motion.

In 1977, the first international chaos conference was held in Italy. It marks the official birth of the science of chaos. The physicist Ford who was one of the assembly moderators considered that chaos theory was the third revolution of physics following relativity theory and quantum mechanics in the twentieth century. He said, relativity theory eliminated illusions about absolute space and time, and quantum mechanics broke Newton’s dream about controllable measurement process; and chaos burst the bubble of determinism of predictability by Laplace [1].

1.2Development of Chaos

In the latter half of the twentieth century, nonlinear science has been developing rapidly, which accounted for a great proportion of the study on chaos. Chaos, as a remarkable frontier project and academic hot topic, reveals the unity of simplicity and complexity, order and disorder, and certainty and randomness in nature and human society, and broadens people’s view greatly, and deepens people’s understanding of the objective world. In the field of natural science and social science, chaos is striking and changing the field of science and technology, and it presents a huge challenge to people.

In the early 1960s, scientists began to explore some elusive phenomenon in nature. In 1963, Lorenz proposed deterministic nonperiodic flow model [2], and later he proposed the butterfly effect. In 1975, Li and Yorke published “Period three implies chaos” [3] in American Mathematical Monthly, which revealed the evolution process from order to chaos. This famous article also defined “chaos” as a new scientific term that officially appeared in the literature. In 1976, American biologist May published “Simple mathematical models with very complicated dynamics” [4] in Nature. This article pointed out that a very simple one-dimensional iterative map can produce complex period-doubling and chaotic motion. He revealed to the people that some simple deterministic mathematical model in ecology can generate chaotic behavior. In 1978 and 1979, Feigenbaum independently discovered scaling property and universal constant [5, 6] in the phenomenon of period-doubling bifurcation, which lays a solid theoretical foundation for chaos in modern science. Lorenz defined that chaos is a science that uses fractal geometry to analyze and study the nonlinear dynamics problem from butterfly effect, aperiodicity, and so on.

In the 1980s, chaos science has been further developed. Many researchers have focused on how an order system turns into the chaotic state and on the characteristics of chaos. In 1981, Takens proposed experimental method of determining strange attractors [7]. In 1983, Glass published an article to set off a wave of computing time series dimension [8]. In 1986, Grassber proposed dynamical systems theory and methods of reconstruction [9]. By calculating chaotic characteristics, such as fractal dimension, Lyapunov exponent from the time series, chaos theory entered into the stage of practical application.

In the 1990s, chaos science got mutual infiltration, promotion, and wide application with other subjects, such as synchronization, secure communication, chaos cryptography, chaotic neural networks, and chaotic economics research areas. Exploring the unique nature, role, and function of chaotic systems to benefit humanity becomes a very great task.

Research on deterministic chaotic system has undergone three stages. The first stage is to study how an order system turns into chaos, which focuses on the conditions, mechanisms, and generated pathways. The second stage is to study what is the order of chaotic system, which focuses on universality, statistical characteristics, and the fractal structure in chaos. The third stage is to study how a chaotic system turns into an order system, which focuses on controlling chaos to achieve order. The current study of chaos focuses on how to control and utilize chaos.

Due to being sensitive on initial conditions, even two identical chaotic systems evolve from almost the same initial conditions, and their orbits also quickly become uncorrelated over time, which makes the chaotic signal become unpredictable and with long-term anti-interception capability. At the same time, chaotic system is deterministic, which is completely determined by the equations, nonlinear system parameters, and initial conditions, and therefore chaotic signal is easy to be generated and replicated. Due to these features, chaos has excellent prospects in secure communications and becomes a hot research topic in chaotic application.

Internationally, the study of secure communications originated in the early 1990s, and it becomes one of the central issues in the field of information science. In 1990, Pecora and Carroll proposed a chaos synchronization method and first observed the phenomenon of chaotic synchronization [10, 11] on electronic circuits. In the same year, Ott, Grebogi, and Yorke realized the control to the chaotic attractor unstable periodic orbits [12]. These works have greatly aroused people’s interest in chaos mechanism and applications, and quickly pushed the theoretical and experimental study on chaos synchronization and control of chaos, and opened the prelude to apply chaos in engineering field. In fact, chaotic encrypted communication will play an important role not only in the field of military, national security, and communications countermeasure, but also in the business world and people’s lives, which is the new trend of future information security technology.

1.3Famous Scientists and Important Events

1.3.1Lorenz and Butterfly Effect

Edward Norton Lorenz (1917–2008) was born in Hartford, Connecticut, USA. He loved science from a young age and graduated from Harvard University in 1940. During World War II, Lorenz served as a forecaster in the United States Army Air Corps. After the war, he received his master’s and doctorate in meteorology. Then he taught at the Massachusetts Institute of Technology. In 1972, he put forward the famous “butterfly effect” theory. In 1975, he became an academician in the US National Academy of Sciences. He won the “Crafoord Prize,” which is called Ecology “Nobel Prize” in 1983 and won the “Kyoto Prize” in 1991. The judges thought his chaos theory “brought the most dramatic change to the human nature view after Newton’s theory.”

The butterfly effect is one of his important scientific contributions, and its discovery is interesting and enlightening. In 1961 winter, Lorenz used computer for numerical prediction calculation. In order to save time, he selected a row data (equivalent to one day’s weather conditions) from the original calculation results as an initial value to calculate again, and then left his office to drink a cup of coffee. After an hour, when he returned to the laboratory, the computer had already calculated the predicted results of the next two months. He was shocked by the new results, which were quite different from the original results. Minor deviation doubled every four days until the similarity between the old and the new data was completely lost. The difference of initial value was less than 1/1,000, but it causes the second calculation result to be completely different from that of the first one. This is the “butterfly effect,” which shows the initial value sensitivity.

In 1979, Lorenz delivered a speech entitled “Butterfly Effect” on the 139th meeting of the American Academy of Science Development and proposed a seemingly preposterous assertion: a butterfly flapping its wings in Brazil, there may be a tornado in Texas, United States, a month later. Thus, he pointed out that it is very difficult to forecast weather accurately. Today, people still talk about this assertion. More importantly, it stimulated people’s interest in chaos. With the rapid development of computer technology, chaos has become a frontier field with far-reaching influence and rapid development. “Butterfly Effect” is fascinating, exciting, and thought provoking, so it not only is a bold imagination and charming color aesthetics, but also lies in the profound scientific connotation and inherent philosophical charm. Lorenz’s discovery laid the foundation for chaos theory. The chaos theory not only affects meteorology, but also profoundly affects every branch of science.

1.3.2Li Tien-Yien and the Concept of Chaos

Li Tien-Yien (1945–) was born in Shaxian, Fujian Province, China. He and his family moved to Taiwan when he was three years old. He received education there until he graduated from university. He is the 68th graduate of mathematics from Tsinghua University, Hsinchu, Taiwan province, China. In 1969, he went to the Department of Mathematics, University of Maryland, United States, for doctoral degree, under Professor York. He was a lecturer in the University of Utah in the United States from 1974 to 1976. Since 1976, he taught at Michigan State University in the United States. Li Tien-Yien has made several important pioneering works and extraordinary achievements in the field of applied mathematics and computational mathematics. The paper “Period Three Implies Chaos” was written by him and York, which first proposed the concept of chaos in science and opened a new era in the scientific community for research in chaotic dynamical systems.

In the past few decades, Professor Li Tien-Yien is suffering from a long-term illness. So far, he has undergone major surgeries under general anesthesia more than ten times and countless surgeries under local anesthesia. Innumerable wounds are on his body. However, he struggled in the face of adversity and overcame the disease with an indomitable spirit of optimism again and again. He sought breakthrough in adversity and fought the disease optimistically. He overcame all the difficulties with a surge of strong determination and ultimately made a first-class research work under difficult environments.

Li Tien-Yien has had rigorous scholarship for decades [13]. He believes that the only way to success is insistence. He said to his students that he was not smart and that it is not really important to be smart. The most important issue is to get to the bottom. He often emphasized to think on the problem just a minute more than others. He believed the precious one minute paves the road for success. He often exhorts his students that with tremendous effort, one must persist till the end of everything and should never give up. He also said that being engaged in researches they must have a thorough understanding, especially on mathematical logic.

Li Tien-Yien witnessed interesting research experience in the birth of chaos. The famous paper, “Period Three Implies Chaos,” also underwent topic selection, research, submission, rejection, and modification, and finally was published in American Mathematical Monthly. It focused on the meaning of section iterative, definitions, and main results of some basic concepts. And some courses are provided as an appendix. A mathematician, who most values clarity and precision, had even talked about mess, which has attracted many readers. Physicists and biologists have also quickly accepted the mathematical articles. Since then, “chaos” which was put forward first by Li and York, has become a technical term in chaos theory and has even become synonymous with chaos theory [14].

1.3.3Feigenbaum and Feigenbaum Constant

Feigenbaum M. J. (1944–), a professor of physics at Cornell University, entered the Graduate School of MIT after graduating from New York College in 1964. He received his PhD in basic particle physics in 1970 and went to Cornell University to engage in postdoctoral research in the same year, but there was no progress for two years. Then he went to Virginia Polytechnic Institute for two years without success. Carruthers P., a former professor in Cornell University, recognized his excellence and selected him as his assistant in Los Alamos (affiliated to California campuses) for free research. Then, after two years, Feigenbaum found a wonderful constant 4.6692. . . [5]. He published the important result in the Journal of Statistical Physics in 1979 [6] and became famous overnight.

Feigenbaum studied about spacing ratio changes in period-doubling bifurcations, which are shown in Table 1.1. He found that bifurcation point of two adjacent spacing Δk forms a geometric sequence with the increase in the number k of bifurcation

1.3.4Leon Ong Chua and Chua’s Circuit

Leon Ong Chua (1936–) is a professor at Department of Electrical Engineering and Computer Science, University of California, Berkeley. He received his bachelor’s degree in electrical engineering in the Philippines Mapúa Institute of Technology in 1959. He received his master’s degree from MIT in 1961 and his doctoral degree from the University of Illinois, Urbana – Champaign in 1964. He taught at the Purdue University in 1964–1970 and became a professor and an IEEE fellow at the University of California, Berkeley, in 1971. He proposed the memristor, Chua’s circuit, and cell-type neural network theory. He is considered the father of the theory of nonlinear circuit analysis. He predicted the existence of the memristor. Thirty-seven years later, the research team of Williams R. S. in Hewlett-Packard developed a solid memristor.

Chua’s circuit provides a hardware foundation for chaos theory and applications research. When Professor Chua was a visiting scholar at Waseda University in 1983, he designed a simple nonlinear electronic circuit, which is shown in Figure 1.2 [15]. The circuit consists of an energy-storage component (capacitors C1 and C2), an inductor L1, an active resistance R, and a nonlinear resistor NR. Nonlinear resistor consists of three linear resistors and an operational amplifier. The circuit can exhibit standard chaotic behavior. Its simple manufacturability makes it an example of chaotic system in real world and becomes a typical model of chaotic systems.

1.3.5Guanrong Chen and Chen’s Attractor

Guanrong Chen (1948–) is a professor in the Department of Electronic Engineering in the City University of Hong Kong. During the Cultural Revolution (for about 10 years), he studied on his own and was directly admitted to the Department of Mathematics of Sun Yat-Sen University at the end of the Cultural Revolution. He graduated in 1981 and received his master’s degree in computational mathematics. Later, he studied appliedmathematics for his doctoral degree in Texas A & MUniversity in 1987. He worked as a visiting assistant professor at the Rice University in the United States from 1987 to 1990, and then as an assistant professor, associate professor, and professor at the Houston University from 1990 to 2000. In 1996, Professor Chen became an IEEE Fellow because of his contributions in chaos control and bifurcation theory and application. Since 2000, he has been serving as a professor of electronic engineering in the City University of Hong Kong and the director of the chaotic and complex network of academic research center. His research focused on related applications areas of control theory of nonlinear systems and dynamics analysis and complex network. In 1999, he proposed a new chaotic attractor [16], known as Chen’s attractor, which is shown in Figure 1.3. It has become an important model in chaos theory and applied research.

1.4Definition and Characteristics of Chaos

Due to the singularity and complexity of chaotic systems, there is no unified definition for chaos. Different definitions reflect the nature of chaotic motion from different angles. The larger impact is the definition of Li–Yorke chaos [3], which is defined from the interval mapping, and it is described as follows.

Li–Yorke theorem: Let f(x) be a continuous map on [a, b]. If there are three cycle points, then for any positive integer n, there are n periodic points for f(x). Li–Yorke chaos definition: For continuous map f(x) on closed interval I, if the following conditions are met, it is confirmed that there is chaos phenomenon: (1) period of periodic points of f(x) is unbounded and (2) the uncountable set S, on closed interval I, meets the following conditions:

Chaos has aperiodic bounded dynamic behaviors in deterministic nonlinear dynamical systems, which is sensitive to initial conditions. Chaotic motion is random-like phenomenon that occurs in deterministic systems. Chaos has the following main features:

1.5Overview of Chaos Synchronization Method

Chaos is so sensitive to initial conditions that people once thought that chaos synchronization is almost impossible. Until 1990, Pecora and Carroll proposed chaos self-synchronization method [10]. Synchronization between two chaotic systems was achieved by using the drive-response method for the first time. This breakthrough research progress broke the traditional idea that movement pattern of chaos is dangerous and uncontrollable. The research results show that chaos not only can achieve control and synchronization, but also can serve as the dynamic basis of information transmission and processing. It is possible to apply chaos to secure communication.

Generally speaking, synchronization belongs to the category of chaos control. Synchronization and control have been combined into one issue by Chua et al. [17]. That is to say, the problem of chaotic synchronization can be considered as a kind of control problem of the chaotic orbit of the controlled system moving according to the orbit of the target system. Traditional chaos control is generally to control a system on unstable periodic orbit, while chaos synchronization is to realize complete reconstruction of two chaotic systems.

1.5.1Chaos Synchronization Methods and Characteristics

Based on chaos synchronization theory and chaos control methods, researchers have put forward many chaos synchronization schemes [18] as shown in Table 1.2.

Drive-Response Synchronization: In 1990, Pacora-Carroll first realized synchronization between two chaotic systems by using drive-response synchronization method. Later, Cuomo and Oppenheim also successfully simulated the synchronization of Lorenz system by using electronic circuit [19]. Soon, Carroll and Pecora promoted drive-response chaos synchronization method to higher-order cascaded chaotic system [20]. The biggest feature of drive-response synchronization is that there is a relationship of drive and response between two nonlinear dynamic systems. Behavior of the response system depends on the drive system, while the behavior of the drive systems has nothing to do with the response system. However, for some actual nonlinear systems, they cannot be decomposed due to physical nature or natural characteristics or other reasons. Then, the drive-response synchronization method will fail.

Table 1.2: Chaos synchronization methods.

Time chaotic synchronization	Spatiotemporal chaos synchronization	Hyperchaos synchronization
Drive-response synchronization	Feedback technology	Drive-response synchronization
Active-passive synchronization	Driven variable feedback	Coupled synchronization
Mutual coupling synchronization	Other	Variable feedback synchronization
Continuous variable feedback		Hybrid synchronization
Adaptive synchronization		Drive and variable feedback
Pulse synchronization		Active-passive feedback
Projective synchronization
D-B synchronization

Active-Passive Synchronization: In 1995, Kocarev and Parlitoz proposed active-passive synchronization method [21]. The advantage of this method is that it can choose the function of drive signal without limitation. Therefore, it is very flexible and has greater universality and practicality, and it includes the drive-response synchronization method as a special case. Active-passive synchronization method is particularly suitable for applications in secure communications. The method can also be used to synchronize control between hyperchaotic systems.

Coupling Synchronization: In 1990, Winful and Rahman theoretically studied the possibility of laser chaos synchronization. In 1994, American scholars Roy and Thornburg and Japan scholars Suga-Wara and Tachikawa independently observed synchronization between two laser chaotic systems in experiments. Study results show that mutually coupled chaotic system can achieve chaotic synchronization under certain conditions. Kapitaniak and Chua achieved synchronization between two Chua’s chaotic circuits with mutual coupling method [22]. Dynamical behaviors of mutually coupled nonlinear systems are very complex. So far there is no universal theory. However, mutually coupled nonlinear systems widely existed in nature, and therefore, it is very important to study this synchronization method.

Continuous Variable Feedback Synchronization: In 1993, German scholars Pyragas and Tamasevicius proposed a method to control continuous nonlinear chaotic systems [23]. Later, this idea was used to study the synchronization between two chaotic systems, and it is called continuous variable feedback synchronization method. Variable feedback synchronization method is simple and easy to implement. According to different systems, the single variable, multivariables, or even all variables of the systemcan be flexibly used to feedback control. Research results show that the minimum number of feedback variables should be equal to the number of positive Lyapunov exponent of the system without perturbation. Meanwhile, multivariable feedback is more effective than single-variable feedback.

Adaptive Synchronization: In 1990, Huberman and Lumer employed adaptive method to control chaos [24]. John and Amritker applied this principle to chaos synchronization, and the phase space trajectory of the chaotic system is synchronized with the desired unstable orbit [25]. Specifically, the adaptive synchronization method is to achieve chaos synchronization by automatically adjusting certain parameters of system with adaptive control technology. There are two prerequisites for the application of this method. (1) At least one or more parameters of the system can be obtained. (2) For the desired orbit, the values of these parameters are known. Adjustment of system parameters depends on two factors: (1) the difference between output variable of the system and corresponding variable of the desired orbit and (2) the difference between the controlled parameter value and the corresponding variable value of desired system.

Pulse Synchronization: In 1997, Yang and Chua put forward pulse synchronization control method [26, 27]. For pulse synchronization, the response system is driven by single pulse transferred from drive signal. The transmitted signal is incomplete chaotic signal, so it is more secure. Pulse synchronization has strong noise immunity and robustness, so it is a promising method of chaos synchronization. According to the basic theory of impulsive differential equations, the stability of synchronization error system depends on its comparative systems. When the comparative system approaches stable state asymptotically, pulses synchronization becomes stable, and the pulse intervals can be obtained.

Projective Synchronization: In 1999, Mainieri and Rehacek observed a new phenomenon – projective synchronization in the research of some linear chaotic systems [28]. By employing this synchronization, for some linear chaotic systems, when an appropriate controller is selected, the output phase of the drive system and response system will be locked, and the amplitude of each corresponding output is also evolved according to a certain proportional relation. This new synchronization method caught researchers’ attention, and a lot of projective synchronization schemes are proposed. Recently, a function projective synchronization method is proposed [29]. Compared with the general projective synchronization, function projective synchronization means that the drive system and response system can be synchronized according to a certain function proportion, which has important significance for the realization of chaotic secure communication. In engineering practice, the parameters of a system may be unknown. In order to realize parameter identification of chaotic systems with unknown parameters, people began to apply adaptive synchronization method to the synchronization research of chaotic systems with unknown parameters. Obviously, combining adaptive control and function projective synchronization to research the synchronization of chaotic systems with unknown parameters has more theoretical value and practical significance [30].

Dead-Beat Synchronization: In 1995, Angeli et al. proposed a synchronization method for discrete chaotic systems, called dead-beat synchronization method [31]. The most important feature of this method is that as long as there are several steps of iteration, chaos synchronization can be achieved accurately. Although several synchronization methods described above can be extended to a discrete system, they are different from dead-beat synchronization method. First, dead-beat chaotic synchronization is accurate synchronization, and other synchronization is asymptotic synchronization in a certain sense. Second, the synchronization time for dead-beat synchronization method is the time for iteration N (N is the dimension of the state space). The synchronization time for other synchronization methods is not only closely related with system parameters, but also depends on the given accuracy range.

Performance of different synchronization methods is different. Based on Lorenz system, researchers have conducted simulation experiments about synchronization time and sensitivity to parameter’s change with the drive-response synchronization, mutual coupling synchronization, feedback perturbation synchronization, adaptive control synchronization, and pulse synchronization. Results show that for feedback synchronization between Lorenz systems, the robustness to parameter changes is the best, and the establishment of synchronization is the fastest. Setup time for adaptive synchronization and pulse synchronization is longer. Weak robustness to parameter change means relatively complex, but its security is better. Therefore, from the synchronization setup time and robustness to parameter changes, the feedback synchronization method is the optimal synchronization method.

1.5.2Other Synchronization Methods and Problems

In the twenty-first century, chaos synchronization study shows a new trend. First, new synchronization methods have been put forward. In addition to improving the existing synchronization methods, the main work is to introduce advanced control theory and technology into chaos synchronization research, such as fuzzy control [32], genetic algorithm [33], and state observer method [34]. All this methods have achieved good results. Second, the object of study has turned to discrete chaotic systems from continuous chaotic system, from low-dimensional chaotic system to high-dimensional hyperchaotic system. Third, how to improve the performance of chaotic synchronization began to draw more attention [35], such as to improve the synchronization system performance with system identification and neural network technology.

Synchronization is the key to achieve chaotic security communication. The most recent and the most competitive studies are to achieve chaotic secure communication based on chaos synchronization. As the foundation of chaos theory and technology, chaos synchronization also is an important aspect of a chaotic mechanism. Many chaos synchronization methods perhaps will not soon get practical value, but each new proposed method will give a new inspiration to open up a new avenue of research. Although many chaos synchronization methods have been proposed, they have not reached the mature stage. There are still many theoretical and technical issues to be resolved in practical applications.

1.6Summary of Chaos Secure Communication

Since 1990, chaotic secure communication and chaotic encryption technology have become a hot topic in the field of international electronic communications [36–39]. So far, the chaos secure communication applications are roughly divided into three categories [18]. The first is to achieve secure communication using chaos directly. The second is to achieve secure communication based on chaotic synchronization signal. The third is digital encrypted communication based on chaos sequence. At present, the second class, chaos synchronous communication, is an international research focus. It has become a new high-tech field. With the development of network technology and computer technology, the third class of chaotic encrypted communication has captured more attention.

So far, four chaos synchronous secure communication technologies have been proposed and developed, including chaotic masking [40], chaos shift keying [41], chaotic modulation [42], and chaotic spread spectrum [43, 44]. The first category belongs to analog communications, and the other three belong to digital communications. At present, the four secure communication schemes are the most competitive technologies. There are three main ways to realize chaos secure communication, such as electrical systems, laser systems, and computer networks. The most mature technology is electrical systems. Here, the overview will focus on the basic principles and study progress of chaotic synchronization secure communication technology based on circuit systems and chaotic encryption technology based on computer network.

1.6.1Chaotic Analog Communication

The typical technique of chaotic analog communication is chaotic masking. Chaotic masking is to superpose the message signal into the chaotic signal directly, and the signal is masked by the randomness of chaotic signal. Cuomo and Oppenheium constructed chaotic masking secure communication system based on Lorenz system [19, 45], combining two response subsystems into one complete response system, which the structure is identical with drive system. The receiver can replicate all states of the sender and achieve synchronization. At the sender, the message signal overlaps with larger amplitude chaotic signal to form noise-like signal. At the receiver, the chaotic signal is obtained from the response system, and the message signal is recovered by subtracting the chaotic signal generated by the response system from the received signal.

Chaotic masking requires that parameters of chaos circuit of the sender and the receiver are accurately matched. Minor differences of parameters may cause synchronization fail. It is very difficult for an attacker to obtain the key parameters, but the strict matching of parameters also puts forward a very high request to the circuit design.

1.6.2Chaotic Digital Communication

Chaos digital communications include chaos shift keying, chaotic parameter modulation, and chaotic spread-spectrum communication. Chaos shift keying is a chaos communication method that is suitable for digital communications. It can code a binary signal using different attractors produced by different parameters. For example, code 1 indicates parameter μ1, and its corresponding chaotic attractor is A1. Code 0 indicates parameter μ2, and its corresponding chaotic attractor is A2. The behaviors of chaos switch between A1 and A2. Response time of the system is controlled by changing the parameter. A parameter at the transmitter is modulated by using a binary signal, and then the chaotic signal is sent to the receiver. The modulated signal can be detected at the receiver by using the synchronization error. Thereby the real signal is recovered. Chaos shift keying has better robustness than that of chaotic masking method, and the anti-interference performance is better, but it has lower information transmission rate. The original signal may be recovered by using the short-time zero crossing rate method, but the security will be reduced. So researchers have proposed an improved chaos shift keying digital communication schemes, including chaotic on-off keying (COOK), differential chaos shift keying (DCSK), frequency modulation differential chaos shift keying (FM-DCSK) modulation, etc. [46–48].

Chaos parameter modulation is to modulate certain parameters of chaotic system by using the original signal in chaos oscillator in the sender. The changes of this parameter reflect changes of the original signal. There is a synchronous chaotic system in the receiver. Because the changes of chaotic state also include the changes in the original signal, the original signal can be detected and recovered by a nonlinear filter. Yang and Chua proposed a chaotic parameter modulation scheme in 1996, which is suitable for general signal modulation [42] and comprehensively analyzed several methods for parameter modulation based on Chua circuit. The simulation results show that, when modulating the different parameters of Chua circuit, the signal recovery accuracy is different. If we modulate capacitor and resistor, the recovery accuracy is higher.

Chaotic spread-spectrum communication is to achieve spread-spectrum communication by using chaotic sequence instead of spreading codes. The traditional code division multiple access (CDMA) technology is mainly limited by the periodicity of the PN code and the available number of orthogonal PN code address. Because chaotic systems are sensitive to initial conditions, a large number of spreading sequences with good correlation properties can be generated through the evolution of chaotic system. Based on the statistical characteristics of the chaotic sequence, it is very effective for the realization of the chaotic spread-spectrum code division multiple access communication. Chaotic sequence can be generated by an initial value and a mapping formula, without having to store values of the sequence. Therefore, it is more suitable for spreading code in spread-spectrum communication, and it has broad application prospects.

In all, chaotic communication has the following advantages [18]:

Based on the advantages above, applications of chaos in secure communication are still in the initial stage of the study; it has captured a lot of attention and interest from physics, information science, and other interdisciplinary fields. It is foreseeable that chaotic secure communication will play an important role in people’s lives, especially in the military and national information security.

1.6.3Encryption Communication based on Chaotic Sequence

In 1989, British mathematician Matthews proposed a new encryption method – chaotic encryption method [49], and thus the study of chaos cryptography began. In the past decades, along with the deepening theoretical study of chaos, the application scope of chaos theory is also expanding. Chaos cryptography application has become a hot topic, and a number of chaotic encryption algorithms have been proposed [50, 51]. Chaotic systems can provide pseudorandom sequence with good randomness, correlation, and complexity. The key sequences generated by a chaotic system not only exhibit excellent cryptographic performances, but also have a rich source. In addition, the chaotic encryption method greatly simplifies the design process of traditional sequence cipher. Those attractive features make it possible to be a stream cryptosystem. The unique advantages of chaos in cryptography attracted many scholars and academic communities around the world to do research.

Unlike chaos analog communication, the anti-crack ability of chaotic encrypted communication is strong. Chaos analog communication using small signal modulation cannot resist the attacks of neural network and regression map. While for chaos encrypted communications, the attacker cannot get the complete original data, and the chaotic signals generated by original equation or regression map reconstruction cannot be obtained according to neural network or regression method. It is difficult to decipher the encrypted signals and it can be applied to highly confidential communication systems.

With the development of computer technology, digital technology, and network technology, chaos encryption software has been emerging persistently [52]. Chaosbased engineering applications have also appeared [53], and the hardware achievements based on Digital Signal Processing (DSP) and Field Programmable Gate Array (FPGA) technologies have also been reported [54]. Chaotic encryption with digital technology has many advantages, such as low requirements for accuracy of circuit elements, easy hardware implementation, easy computer processing, less loss of information transmission, versatility, and so on. Especially when it is used in real-time signal processing, the encrypted information is difficult to be deciphered. Therefore, chaotic encryption shows strong vitality in secure communications.

Compared with conventional packet encryption system, chaos encryption system has the following characteristics:

1.7Chaos Research Methods and Main Research Contents

Because of the complexity of chaos, the research on the chaotic motion should employ many methods, including theoretical analysis, numerical simulation, experimental study, and so on. At present, the methods on the chaos theory mainly include numerical calculation, symbolic dynamics, Melnikov and Shilikov method, phase space reconstruction method, and so on. The numerical methods for judging the chaotic state are the Lyapunov exponents, fractal dimension, power spectrum, Poincaré section, direct observation, and so on. Among them, the Lyapunov exponents and fractal dimension are quantitative methods, and the others are qualitative descriptions. Power spectral density shows the signal changes with the frequency by Fourier transform. The power spectrum of a chaotic signal is changed continuously. Poincaré section is a section in the phase space. The phase trajectory leaves a point at the cross section when it passes the cross section so that the flow in n-dimension can be reduced to a map with n – 1 dimension, and many properties of the original system are preserved. For a chaotic system, these points have a fractal structure and cannot fill up the entire section. To observe whether there exists butterfly effect is a typical direct observation method. Comparing the time series of chaotic system, two very similar initial values after a short-time evolution will be very different. It reflects the sensitivity of chaotic system to initial value. For the chaotic systems with known dynamics equations, the symbolic dynamics method is used to study the dynamics equation. In the second chapter of this book, we will focus on the analysis methods of chaotic characteristics.

Numerical simulation is an important method for the study of chaos theory and application. First of all, numerical simulation method provides the application issues associated with the real system for chaos theory analysis, and it broadens the field of theory research. Simulation system provides ideal plasticity experimental model for theoretical research, and it opens up the new field of chaotic application study. Second, the simulation method can provide the basis for the feasibility of constructing the real and practical system. The third and fourth chapters of this book are based on the Matlab/Simulink numerical simulation method. The third, fourth, and fifth chapters mainly use the numerical simulation method based on Matlab/Simulink to study.

Real experimental study is an important method for the practical application of chaos. Since the 1990s, the results of a large number of theoretical studies are based on real system experiments and applications. Real experiments include circuit experiments and computer experiments. With the development of microelectronics technology, it is possible to design various kinds of high precision chaotic components and systems. The development of network technology and computer technology provides a platform for constructing and testing various chaotic encryption softwares. The fifth and sixth chapters of this book will be mainly based on computer system and network.

Three kinds of chaos analysis methods discussed above are mutual dependence and mutual promotion, and constantly push forward the development of chaos research. In this book, the theoretical analysis, numerical simulation, and real experiments are combined. This book will mainly use the Lyapunov exponents, conditional Lyapunov exponents, and fractal dimension for the theoretical study. Based on Matlab/Simulink simulation platform, we analyze chaotic system synchronization control mechanism and the chaotic synchronization performances. Through computer advanced language and computer network, we carry out real experimental researches, designing practical chaotic encryption software, testing it, and analyzing the security.

In the study of chaos synchronization control, both the continuous system synchronization and discrete system synchronization are studied. The relationship between the performance and the control parameters is particularly studied. In the application of chaotic systems, we will focus on the integration of chaotic dynamics, advanced control technology, modern cryptography, and modern communication technology. Combining the physical synchronization of chaotic system with computer programming technology, static data file encryption softwares are designed. The network-based self-synchronization method is investigated. Chaotic encryption software to achieve real-time voice signal transmission is designed by using the advanced computer language. Those above are the main achievements of the research group in the field of chaotic application.

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