Memristors and networks: new structures for complexity

Abstract

One of the most interesting aspects of complexity is that it occurs at different levels. It may occur at the level of interactions among the agents that compose a complex network: despite the relatively simple behavior of each single unit, the whole network may exhibit holistic collective dynamics, such self-organization, synchronization, robustness to failure, and so on; and it may occur, in the form of an aperiodic irregular behavior, at the level of a system described by a low-order set of ordinary differential equations, three, for instance, in the case of continuous-time systems. This thesis focuses on both levels of complexity.

The first part, in particular, deals with complexity at the level of a single dynamical system. The main contributions of the work summarized in this thesis refer to the use of a new electronic component for the design of chaotic circuits. This new component, the memristor, is at the same time a memory element and a nonlinear element and for this reason has been regarded in literature as an effective block to reduce the minimum number of components needed to build a chaotic circuit. The original aspect of this thesis is the focus on a realistic model of memristor, that is a model derived starting from the analysis of the real memristor device discovered in the HP laboratories. The use of such approach introduces constraints in the design that are not considered in idealized models such as piece-wise linear ones. The main results were: i) the introduction of a configuration of two memristors in antiparallel which has been used as the fundamental block to design a gallery of autonomous and non-autonomous nonlinear circuits exhibiting a rich dynamics, including chaos; ii) the design of a hybrid circuit which takes from the characterization methodology of real memristors the idea of using a simple digital linear control circuitry which allows chaos to be observed with the driving of a single memristor.

The second part of the thesis focuses on synchronization on complex networks. In particular, the onset of a new form of synchronization, named remote synchronization, in complex networks has been investigated. Remote synchronization appears in star-like networks of coupled Stuart-Landau oscillators, where the hub node is characterized by an oscillation frequency different from that of the leaves, as a regime in which the peripheral nodes are synchronized each other but not with the hub. In this thesis we have investigated if similar conditions can be observed in more general frameworks. We have found that networks of not homogeneous nodes may display many pairs of nodes that, despite the fact that are not directly connected nor connected through chains of synchronized nodes, are phase synchronized. We have introduced measures to characterize this phenomenon and found that it is common both in scale-free and Erdos-Renyi networks. Furthermore, this is an important mechanism to form clusters of synchronized nodes in a network. Finally, we have linked the appearance of pairs of remotely synchronized nodes to a topological condition of inhibition of direct paths or paths through chains of synchronized nodes, thus elucidating a mechanism which has lead to the definition of a series of topologies where remote synchronization is found.

Finally, we have explored the use of memristor as a synapse for complex networks. Also in this case, we have used a configuration of two HP memristors and shown that such configuration provides an adaptation rule for the links of a complex network, enabling the emergence of a set of weights leading to synchronization.