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Chapter 1
Introduction: Dynamical Chaos and Information Communication
The discovery of dynamical chaos oscillations of deterministic origin possessing the properties of random processes, such as continuous spectrum, finite correlation time, high sensitivity to perturbations, unpredictable behavior on large time intervals — the brightest event in nonlinear science in the recent decades. The considered oscillations are self-oscillatory motions of nonlinear systems and their properties are fully determined by the properties of the dynamical system. The understanding that complex noise-like oscillations (dynamical chaos) may arise in nonlinear systems at zero random action changed the traditional conception of oscillatory and wave phenomena dramatically. It became clear that chaotic oscillations of one or another degree of chaoticity rather than simple periodic oscillations are typical for the majority of physical, chemical, biological and other natural systems. The discovery of dynamical chaos aroused great interest in the scientific community and has been a challenging field of research since then.
The pioneering works on dynamical chaos were published at the beginning of the 1960s and initiated theoretical and experimental studies of this phenomenon in many laboratories worldwide. The basic concepts related to dynamical chaos were formulated by the beginning of the 1990s [4–11].
Investigation of the principal properties of dynamical chaos stimulated an interest in the application of this phenomenon in engineering systems that would use the features of dynamical chaos. One of the promising trends was to use chaos in communication systems. Dynamical chaos possesses many attractive properties that may be useful for information transmission [12]:
•a possibility of producing complex oscillations by means of devices with a simple structure;
•realization of a great variety of chaotic modes in one device;
•controlling chaotic modes by small variations of system parameters;
•large information capacity;
•diversity of methods of introducing information signal into a chaotic one;
•higher modulation velocity as compared to regular signals;
•novel methods of multiplexing;
•secure communication.
The trend of using dynamical chaos for communications started at the beginning of the 1990s when quite a few chaotic oscillators were created, such as inertial nonlinearity oscillator [13], ring self-excited oscillator [9], tunnel diode oscillator, Chua’s circuit [7], and the phenomena of chaotic synchronization [14–18] and chaotic synchronous response [19] were discovered.
Beginning in 1992, some methods of information communication using chaotic signals were proposed: chaotic masking, chaos shift keying, nonlinear mixing, etc. They demonstrated a possibility of using chaos for data communication, thus creating the background for the development of the new trend in communication systems.
Early experiments on information communication [20–25] confirmed that it is possible, in principle, to transmit information using chaos. However, development of this trend faced severe problems. Firstly, communication circuits based on self-synchronization of chaotic oscillations are very sensitive to distortions in the channel, noises and incomplete identity of transmitter and receiver parameters. Secondly, in the proposed circuits, chaos was used as undercarrying oscillations modulating a highfrequency carrier. As a result, the attractive feature of chaos, that is, its broadband that was intended to ensure high-speed communication of signals with large data base, was lost. Consequently, further development of chaotic communication systems demanded solutions for a number of topical problems, which included elaboration of high-efficiency generators of chaos operating in a straightforward manner in a wide frequency range covering high and ultrahigh frequencies, and creation of methods that would ensure stable synchronization of chaotic oscillations.
It seems promising to employ phase and frequency systems as a solution to these problems [26]. Information communication devices in which regular signals are used are known to broadly employ phase-locked loops (PLL) and frequency locked loops (FLL). These systems were initially developed for solving the problems of synchronization, frequency stabilization, controlling the frequency and phase of radio oscillations, filtration, demodulation, signal formation and processing, and some other problems. High accuracy, reliability, noise stability, controllability, capability of operating at high and ultrahigh frequencies, and technological effectiveness made these systems an essential part of nearly all communication systems. Naturally, the features enumerated above make such systems attractive for creation of novel communication systems using chaotic signals instead of the traditional regular ones.
At present, the phase and frequency systems for regular signals have a well-developed theory [27–32], whereas the application of PLL and FLL for information communication using chaotic signals still lacks theoretical basis. The results available in the theory of dynamical behavior of such systems in asynchronous modes are insufficient for explaining numerous phenomena occurring outside the domains of existence and stability of the synchronous regime (including domains of existence of self-modulation modes), for defining dynamical characteristics of chaotic signals generated by these systems, as well as for goal-oriented development of communication systems using chaotic modes of PLL and FLL.
A specific feature of the considered class of systems is the presence of phase or frequency control circuits, allowing the frequency of the controlled oscillators to be stabilized relative to the regular reference signals in a wide range of initial deviation of frequency. However, outside the synchronization domain, these circuits provide a rich potential to excite various self-modulation oscillations, including the chaotic ones. It should be emphasized that the chaotic signals formed at the output of controlled PLL oscillators may be transmitted to the communication channel immediately after formation, without additional transformations, which is an undoubted merit of these systems. Another advantage of the considered systems is their ready integration into an ensemble by means of different couplings between PLL and FLL systems. The new properties arising as a result of such integration expand the functional potential of the systems, both as traditional frequency controlled generators of periodic signals and generators of chaotic signals. In practice, the considered systems are integrated in an ensemble when it is necessary to meet conflicting requirements to different characteristics: pull-in band, filtering properties, speed, cycle skip probability, etc. [28, 30–32]. There are also problems when several systems should be unified in an ensemble [4], for instance, optimal reception and assessing parameters of complex signals. As for new applications, particularly for chaotic communication systems, of major interest are complex chaotic self-modulation oscillations that are realized in such ensembles. Note that dynamical properties of self-oscillatory systems are determined not only by parameters of the systems but also by the structure and force of coupling between the systems, thus allowing coupling parameters to be used as control ones.
Note that the considered ensembles of coupled voltage controlled oscillators are a variety of multielement self-oscillatory systems that currently attract the attention not only of physicists but also of biologists, chemists, economists, and so on [4]. The nonlinear phenomena of collective dynamics demonstrated by such systems (synchronization processes, self-oscillatory regular and chaotic modes) are, firstly, important for ascertaining the basic regularities of dynamical behavior of coupled frequency and phase-controlled oscillators and, secondly, they may be useful for investigation of other objects (multielement-phased arrays, Josephson contacts, power circuits, spatio-temporal processing systems, etc.).
The book consists of eight chapters. The first chapter is introduction. The second chapter is concerned with the investigation of the nonlinear dynamics of a typical model of PLL system with second-order filter described by a third-order differential equation defined in cylindrical phase space. The solutions to the model are analyzed using the mathematical apparatus of phase space. The correspondence between attractors of the mathematical model and real dynamical modes of the PLL system is established. These dynamical modes may be classified into three groups: synchronous, quasisynchronous and asynchronous. The quasisynchronous PLL modes are especially interesting for the formation of chaotic oscillations with angular modulation and stabilized carrier frequency. In the study of the nonlinear dynamics of the PLL model, primary attention is focused on the investigation of methods to achieve quasisynchronous modes and their chaotization processes, analysis of the properties of chaotic modes, and study of the structure of regions of existence of quasisynchronous modes in the space of parameters.
Chapters 3–5 address the problems of nonlinear dynamics of small ensembles of PLLs. Models of two and three cascade-coupled PLLs are studied in Chapters 3 and 4, and features of behavior of an ensemble of two PLLs coupled in parallel are considered in Chapter 5. Dynamical properties of the ensembles are investigated by analyzing trajectories in the phase spaces of the corresponding mathematical models described by systems of ordinary differential equations. The models of the phase systems are defined in cylindrical phase spaces containing several angular coordinates. This significantly complicates the study of the dynamics of phase-controlled oscillators as the considered models have a greater variety of motions than those defined in Cartesian coordinates. It is shown that the integration of phase-controlled oscillators in an ensemble results in changes in the characteristics of the dynamical modes of partial systems and in the appearance of new modes that are not typical of partial systems. Scenarios of the onset of self-oscillatory modes, mechanisms of their chaotization, and domains of existence of chaotic oscillations in the space of parameters are investigated. The properties of chaotic oscillations are analyzed as a function of parameters of phase systems and coupling parameters. It is shown that the properties of the generated oscillations may be controlled quite efficiently by small changes in coupling parameters. Results of physical experiments with an ensemble of two cascade-coupled phase systems are presented. These experiments confirm the results of the theoretical studies and indicate that PLL ensembles are effective generators of chaotically modulated oscillations in a wide region of parameter space.
Problems concerned with synchronization of chaotically modulated oscillations of two unidirectionally coupled systems, both single and belonging to an ensemble, are addressed in Chapter 6. For synchronization of chaotic oscillations of phase-controlled oscillators with non-identical but close parameters, it is proposed to use the principle of automatic tuning (synchronization). It is demonstrated that by following this principle it is possible to synchronize chaotically modulated oscillations to a rather high accuracy and to broaden existence domains of the mode of chaotic synchronization in the space of parameters for both single-loop oscillators and oscillators combined in an ensemble.
Results of computer simulations of information communication using dynamical chaos are presented in Chapter 7. The experimental results lead to a conclusion that it is promising to construct new communication systems using dynamical chaos. Chapter 8 is the conclusion.
A few comments on the adopted terminology: The main topic of this book is collective dynamics of coupled phase-locked loops (PLL). In the literature, such systems are also frequently referred to as phase synchronized systems (PSS). Sometimes, they are called phase-controlled systems or simply phase systems (PS). The terms PLL, PSS and PS are used in the text as synonyms.
The term synchronization of chaotic oscillations is rather ambiguous in the literature and greatly differs from the term for regular periodic oscillations. To avoid ambiguity, we give some comments on the use of this term in Chapter 5 that is devoted to the synchronization effects.
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Chapter 2
Nonlinear Dynamics of the Phase System
2.1.Mathematical Models
The continuous system of phase-locked loops (PLL) is a typical ring of the automatic frequency control generator. Figure 2.1 (see [28]) shows the structural scheme of the continuous system of PLL. The basic elements of the ring are as follows: the controlled generator (G) (or voltage controlled oscillator), the phase discriminator (PD), the low frequency filter (F) and the control element (CE). The scheme functions according to the following principle. The periodical oscillations from the output of the generator G with the current ...
