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Fortunately an important feature in our living world is the tendency to achieve common rhythms of mutual behavior, or is, in other words, the tendency to synchronization . This phenomenon of synchronization is extremely wide spread in nature as well as in the realm of technology and society. Synchronization is usually understood as the capacity of objects of different nature to form a common operation regime due to interaction or forcing. The fact that various objects seek to achieve order and harmony in their behavior, which is a characteristic of synchronization, is a manifestation of the natural tendency to self-organization existing everywhere in nature [1–5, 7–10].

Synchronization phenomena studied in this book are mainly based on the dynamics of phases and frequencies of oscillations. Because there is no general way to introduce a phase and a frequency for arbitrary oscillatory system, we will present in this chapter several often used phase and frequency definitions. We start with classical definitions of phase and frequency of oscillations introduced for the simplest case, the harmonic oscillator (Sect. 2.1).

In this chapter we describe synchronization phenomena of periodic and chaotic oscillators subject to an external periodic force (Fig. 3.1). In the classical theory of synchronization the periodically driven oscillator is the main and historically first studied model. The electrical engineers Appleton [102] and van der Pol [48] were the first to show the possibility of synchronization of triode generator by a weak external periodic signal. Then, external synchronization of self-oscillatory systems was theoretically studied by the Russian physicists Andronov and Vitt [103, 104], Mandelshtam and Papaleksi [105].

In this chapter we go the next step and analyze synchronization phenomena in systems of two coupled elements. We start (Sect. 4.1) with the classical case, i.e., two coupled regular systems: We analyze two weakly coupled arbitrary limit-cycle oscillators (Sect. 4.1.1), the nonscalar coupled van der Pol oscillators (Sect. 4.1.2) and finally two coupled active rotators (Sect. 4.1.3). Then (Sect. 4.2), synchronization of coupled chaotic systems is presented. Starting with the analysis of chaotic Rössler resp. intermittent chaotic oscillators (Sects. 4.2.1 and 4.2.2), Sect. 4.2.3 is devoted to the investigation of coupled chaotic phase oscillators. Finally (Sect. 4.3) we treat two coupled circle maps as a prototypical discrete in time system.

In this chapter we are starting with the main part of this book, the treatment of synchronization phenomena in ensembles or networks of coupled oscillators. First we treat networks of coupled first-order phase oscillators. This choice is based on the fact that coupled phase oscillators are a basic model for analyzing synchronization processes in large ensembles of oscillatory systems: limit-cycle oscillators (see Chap. 6) and chaotic oscillators (see Chap. 7).

In this chapter collective effects in chains of diffusively coupled limit-cycle oscillators with different individual frequencies are investigated by asymptotic and numerical methods. In the case of weak coupling using phase approximation (see Chap. 5), conditions for the onset and existence of a global synchronization regime are determined. Cluster synchronization regime and mechanisms of synchronization transitions are studied. Main results are (1) the existence of mono- and multistable regimes of cluster synchronization and (2) the soft respective hard transitions between these structures, which are consisting of a different number of clusters. Synchronization is observed in a broader range of parameters in a randomly formed chain than in the case of a regular arrangement of oscillators with monotonically varying individual frequencies along the chain.

Synchronization in ensembles of coupled chaotic oscillators is a topic of great interest, due to its high theoretical significance and many applications in a variety of fields including neural networks [250–256], electronic circuits [257–259], optics [260–264], chemistry [67, 265, 266], etc.

In the previous chapters synchronization phenomena in ensembles of time-continuous oscillators are treated. Many systems in nature and technology and their corresponding mathematical models are, however, discrete in time, e.g., population dynamics [285, 286], asteroidal motion [287], many systems under external force [288, 289], laser dynamics [290]. Because of that it is important to consider synchronization effects in ensembles of coupled time-discrete elements.

In this chapter we study conditions for an onset of regular and chaotic phase synchronization (PS) regimes in ensembles of coupled circle maps (CMs) [183]. For networks of coupled maps different problems of synchronization, pattern formation, and spatiotemporal chaos have been investigated [306–311].

In this chapter we present an automatic control method of phase locking of regular and chaotic nonidentical oscillations, when all subsystems interact by a feedback [332]. This method is basing on the well-known principle of feedback control which takes place in nature and is successfully used in engineering. Considering the models of coupled systems in biology, neuroscience, and ecology one can see that in many of them the coupling between interacting elements is nonlinear , and usually has the form of quadratic functions of the subsystem variables. Such a coupling serves as the basis of an internal self-organization mechanism leading to a balanced motion in these systems. Synaptically coupled neurons [342, 343], phase transitions in human hand movement [344], ecological systems [339], or spinal generators of locomotion [345], are only some well-known examples of balanced cooperative oscillatory motion, caused by such a nonlinear coupling. In engineering, nonlinear coupling, is used, for example, in coupled lasers [333, 334] or phase-locked loops (PLL) [57] (see also Sect. 2.5).