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The brain can be described as a very complex dynamical system whose autonomous, intrinsic activity is modulated by a great variety of external inputs. In the last decade, the simulation of this activity by using networks of neurons has led to the development of new approaches to describe the processing of information by the nervous system. In this review paper we focus on winnerless competition, a new concept allowing the analysis of the emergent behavior associated with collective synchronization, multi-stability and adaptation — properties which seem to be at the basis of brain performance. The relationship between winnerless competition and the concept of chaotic itinerancy is stressed by pointing out their common association with the formation of complex heteroclinic orbits in the high-dimensional phase space associated with the dynamics of complex neuronal systems. The potentialities of networks of coupled map neurons to the study of activation and synchronization regimes in neural systems are suggested by means of different simulations showing a great variety of dynamical phenomena.

The present paper proposes two types of parametrically coupled circle map networks for multistate associative memory. One of the networks uses a circle map exhibiting an attractor-merging crisis of multiple chaotic attractors to represent a multistate element. The other uses another circle map whose bifurcation diagram serves as a substitute for a multilevel activation function. The configuration of each network is suitably selected according to the dynamics of the individual circle map so that the network can bring about self-organizing chaotic dynamics with an association of a memory. Namely, the coupling term is determined by the generalized partial error function in the first network, and by the weighted sum of inputs in the second network. These multistate networks can be considered as extensions of two kinds of interesting binary networks called the parametrically coupled sine map networks [Lee & Farhat, 2001a], respectively. We illustrate that the proposed networks can exhibit desirable associative dynamics that is missing in the conventional multistate networks.

We study the dynamical behavior of the collective field of chaotic systems on small world lattices. Coupled neuronal systems as well as coupled logistic maps are investigated. We observe that significant changes in dynamical properties occur only at a reasonably high strength of nonlocal coupling. Further, spectral features, such as signal-to-noise ratio (SNR), change monotonically with respect to the fraction of random rewiring, i.e. there is no optimal value of the rewiring fraction for which spectral properties are most pronounced. We also observe that for small rewiring, results are similar to those obtained by adding small noise.

We introduce a new method for determining the global stability of synchronization in systems of coupled identical maps. The method is based on the study of invariant measures. Besides the simplest nontrivial example, namely two symmetrically coupled tent maps, we also treat the case of two asymmetrically coupled tent maps as well as a globally coupled network. Our main result is the identification of the precise value of the coupling parameter where the synchronizing and desynchronizing transitions take place.

Coupled Map Lattices (CML) are a kind of dynamical systems that appear naturally in some contexts, like the discretization of partial differential equations, and as a simple model of coupling between nonlinear systems. The coupling creates new and rich properties, that has been the object of intense investigation during the last decades. In this work we have two goals: first we give a nontechnical introduction to the theory of invariant measures and equilibrium in dynamics (with analogies with equilibrium in statistical mechanics) because we believe that sometimes a lot of interesting problems on the interface between physics and mathematics are not being developed simply due to the lack of a common language. Our second goal is to make a small contribution to the theory of equilibrium states for CML. More specifically, we show that a certain family of Coupled Map Lattices presents different asymptotic behaviors when some parameters (including coupling) are changed. The goal is to show that we start in a configuration with infinitely many different measures and, with a slight change in coupling, get an asymptotic state with only one measure describing the dynamics of most orbits coupling. Associating a symbolic dynamics with symbols +1, 0 and -1 to the system we describe a transition characterized by an asymptotic state composed only of symbols +1 or only of symbols -1. We rigorously prove our assertions and provide numerical experiments with two goals: first, as illustration of our rigorous results and second, to motivate some conjectures concerning the problem and some of its possible variations.

Network architecture can lead to robust synchrony in coupled maps and to codimension one bifurcations from synchronous fixed-points at which the associated Jacobian is nilpotent.

We discuss the codimension one synchrony-breaking period-doubling bifurcations for three-cell coupled maps. Interesting phenomena occur for all these coupled maps — a branch of period-2 points with amplitude growing as |λ|^⅙ for coupled networks of feed-forward type, as well as multiple (two) branches of period-2 points with amplitude growing as |λ|^½ for coupled networks of feed-forward type.

We also discuss how some results related to patterns of synchrony that are valid for coupled vector fields are also valid for coupled maps.

A two-dimensional parametrically forced system constructed from two identical one-dimensional subsystems, whose parameters are forced into periodic varying, with mutually influencing coupling is proposed. We investigate bifurcations and basins in the parametrically forced system when logistic map is used for the one-dimensional subsystem. On a parameter plane, crossroad areas centered at fold cusp points for several orders are detected. From the investigation, a foliated bifurcation structure is drawn, and existence domains of stable order cycles with synchronization or without synchronization are detected. Moreover, evolution of bifurcation curves with respect to a coupling intensity is analyzed. Basin bifurcations and preimages with respect to critical curves are described. Basins where boundary depends on the invariant manifold of saddle points are numerically analyzed by considering second order iteration and using superposition with Newton method, although the system has discontinuity regarding parameters.

The paper examines the appearance of on-off intermittency and riddled basins of attraction in a system of two coupled one-dimensional maps, each displaying type-III intermittency. The bifurcation curves for the transverse destablilization of low periodic orbits embeded in the synchronized chaotic state are obtained. Different types of riddling bifurcation are discussed, and we show how the existence of an absorbing area inside the basin of attraction can account for the distinction between local and global riddling as well as for the distinction between hysteric and non-hysteric blowout. We also discuss the role of the so-called mixed absorbing area that exists immediately after a soft riddling bifurcation. Finally, we study the on-off intermittency that is observed after a non-hysteric blowout bifurcaton.

The field of environmental sciences is abundant with various interfaces and is the right place for the application of new fundamental approaches leading towards a better understanding of environmental phenomena. For example, following the definition of environmental interface by Mihailovic and Balaž [23], such interface can be placed between: human or animal bodies and surrounding air, aquatic species and water and air around them, and natural or artificially built surfaces (vegetation, ice, snow, barren soil, water, urban communities) and the atmosphere. Complex environmental interface systems are open and hierarchically organised, interactions between their constituent parts are nonlinear, and the interaction with the surrounding environment is noisy. These systems are therefore very sensitive to initial conditions, deterministic external perturbations and random fluctuations always present in nature. The study of noisy non-equilibrium processes is fundamental for modelling the dynamics of environmental interface systems and for understanding the mechanisms of spatio-temporal pattern formation in contemporary environmental sciences, particularly in environmental fluid mechanics. In modelling complex biophysical systems one of the main tasks is to successfully create an operative interface with the external environment. It should provide a robust and prompt translation of the vast diversity of external physical and/or chemical changes into a set of signals, which are “understandable” for an organism. Although the establishment of organisation in any system is of crucial importance for its functioning, it should not be forgotten that in biophysical systems we deal with real-life problems where a number of other conditions should be reached in order to put the system to work. One of them is the proper supply of the system by the energy. Therefore, we will investigate an aspect of dynamics of energy flow based on the energy balance equation. The energy as well as the exchange of biological, chemical and other physical quantities between interacting environmental interfaces can be represented by coupled maps. In this chapter we will address only two illustrative issues important for the modelling of interacting environmental interfaces regarded as complex systems. These are (i) use of algebra for modelling the autonomous establishment of local hierarchies in biophysical systems and (ii) numerical investigation of coupled maps representing exchange of energy, chemical and other relevant biophysical quantities between biophysical entities in their surrounding environment.