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The design of systems capable of storing and processing information is an important problem in modern science and technology. One of the directions intensively developing deals with neuro-inspired information processing systems. This implies designing systems using some operation principles of the nervous system of animals [5.1, 5.3, 5.4, 5.8, 5.21, 5.22]. The nervous system consists of a rich variety of neurons, which are huge in number with a still higher number of connections (synapses). Abstracting from details we may define three main states of a neuron. These are the base state, which can be at rest or in a so-called subthreshold oscillation mode which is a robust quasiharmonic oscillation; the excited state (or more than one as in the Inferior Olive); and the refractory state, where the neurons do not respond to inputs. By means of various connections (electrical or chemical synapses) the neurons form extremely complex spatially distributed neural networks [5.5–7,5.10,5.19,5.20]. It has been argued, and even experimentally established, that storing and processing information in the nervous system of animals is connected with the appearance of spatio-temporal structures of activity in the neural networks. Such structures are formed by the cooperative self-consistent action of neurons, hence self-organization of the network. From the viewpoint of nonlinear dynamics the neural network represents a spatially distributed discrete active system (or discrete active medium).

Hopefully, the reader has found that the results presented in our book give a fair view of self-organization processes together with a great deal of methodology for a very diverse class of active spatially extended systems. The contents of the book may also appear divided into two parts. One deals with selforganization in single systems and the other with processes in interconnected active 1D (chain) and 2D lattices, thus leading to a 3D architecture.

The discovery of the Josephson effect stimulated the development of superconductor microelectronics with a huge number of both theoretical and experimental papers devoted to it. Such interest is caused by its great technological potential and the value of the Josephson effect in information processing, generation of ultra-high frequency oscillators, fast switching devices, mixers, detectors, amplifiers and so on (see, e.g., [3.1–3,3.7]).

In modern science, “synergetics” owes its development to the extraordinary research output provided by H. Haken coupled with the many workshops he has organized in the past three decades. In short it deals with the emergence of dynamical and/or evolutionary features in complex systems where the total is not the mere addition of parts. Hence an emergent property belongs to a higher level of description than that of the underlying elements. Emergent properties have genuine laws of their own as is customary in biology, ecology, sociology, etc., and indeed engineering, chemistry and physics. Already thermodynamics is a science of emergence (of cooperative properties) in underlying atoms, and statistical mechanics is the bridge between the micro and macro (or even meso) levels. Equilibrium critical phenomena and phase transitions provide paradigmatic examples of transitions and cooperativity in nonevolving systems, and they define an area of physics rather well understood both at the emergent level and in the way emergent, macroscopic and phenomenological properties originate from the, microscopic, lower level. It is one of the areas where from first principles we know the recipe to account for emergent, synergetic, properties.The reductionism from the emergent to the lower level is far from trivial and clearly indicates that, although the basics are to be found at the microlevel, understanding the emergence of cooperative properties demands a great deal of imagination and, on occasion, computer power.

Starting with the discovery of deterministic chaos in a 3D dynamical system made by Lorenz in 1963, a great deal of effort was made to build electronic circuits exhibiting chaotic oscillations. At present a large number of such items have been proposed. In particular, special interest has been devoted to the so-called Chua’s circuit (oscillator). Although earlier related proposals existed, this circuit was introduced – in the proper context – by L.O. Chua at the opening lecture given in the Workshop on Nonlinear Theory and its Applications (NOLTA’92), held at Waseda University, Tokyo, in January 1992. Chua’s circuit possesses a large variety of possible dynamical behaviors. By changing the values of control parameters, one can obtain regular behavior or chaotic oscillations. [4.2, 4.6, 4.7]

In this chapter we consider lattice models with discrete time, i.e. dynamical systems with a discrete time, discrete space and continuous state and hence coupled map lattices (CML). CML describe the collective dynamics of a finite or infinite number of interacting finite-dimensional local dynamical systems (elements, units or cells) placed on the sites of a spatial lattice (the “physical” space). The dynamics of a single element is described by a point map. Thus, CML are characterized not only by their internal dynamics, i.e. that of the local elements, but also by the dynamics in the “physical” space. CML have been used to model a variety of phenomena including pattern formation, nonlinear waves, topological spatial chaos, distribution of defects in the spatial structures of nonlinear fields, reentry initiation in coupled parallel fibers, etc. [7.1–3, 7.6, 7.7, 7.9–16, 7.18, 7.20–22, 7.24, 7.25–31].

A solitary wave that, eventually, becomes a soliton represents a unique and very attractive object in modern nonlinear science of spatially extended systems. The term “soliton” was coined by N.J. Zabusky and M.D. Kruskal [2.30] to characterize solitary waves in nonlinear systems whose properties (energy etc.) are mostly localized in a bounded region of space at any instant of time such that upon collision they act like particles (e.g. electron, proton, etc.) and, in the case they studied, elastically, crossing each other or interchanging places with no significant change of form.