Multistability is a phenomenon occurring in dissipative systems when several stable attractors coexist for a given set of the system parameters. This phenomenon has been observed in many fields of science, including electronics (Maurer and Libchaber 1980), optics (Brun et al. 1985; Ruiz-Oliveras and Pisarchik 2009), mechanics (Thompson and Stewart 1986), and biology (Foss et al. 1996). The mechanisms responsible for multistability can be many and diverse, including homoclinic tangencies (in weakly dissipative systems) and delayed feedback.

In spite of the possible differences in the origin of multistability, the behaviors of multistable systems share several similarities. All multistable systems are characterized by an extremely high sensitivity to initial conditions, so that even very tiny perturbations can cause a significant change in the final attractor state. Furthermore, their qualitative behavior often changes drastically under just very small variations of their parameters. Finally, the intervals of coexistence of attractors can be so small that a slight perturbation in a control parameter can cause a rapid change in the number of coexisting attractors, giving rise to very complex dynamics.

Synchronization of multistable chaotic systems is particularly relevant to sociology and brain research, where it helps scientists acquire a better understanding of the mechanisms underlying the formation of social groups, the sudden emergence of new ideas, the adoption of innovations, the mechanisms of decision making, and the spread of habits, fashions, leading opinions, and panic. Considering individuals or groups of people as dynamical units, one could say that coordination in their behavior is a type of synchronization. For example, in a crowd, people lose their individuality and act synchronously as a common system. This occurs very frequently when people are affected by a common external force, for instance when they look at or listen to the same object, be it a political leader, a movie, a football game, or a rock concert.

Psychologists and sociologists have long been studying this phenomenon. The first classic treatment of crowds was offered more than 100 years ago by Gustave LeBon (1896), who interpreted the mob behavior during the French Revolution as an irrational reversion to wild animal emotions. Later, a similar view was expressed by Freud (1922).

The final chapter of this book is devoted to the study of the synchronization properties of systems whose structure of connections forms a complex network. The main focus is to show how the structural properties of the networks affect synchronizability. In addition, we discuss how to assess the stability of the synchronous state in networks, including the case of time-varying structures and multilayer networks. Finally, we describe an experimental setup that can help in studying networked systems.

Introduction

In the previous chapters we introduced distributed systems and started discussing their dynamics. So far, we have dealt with a rather simple connection topology: each system was either coupled to every other system, or only to its nearest neighbors in some spatial arrangement. However, such simplifications are not always adequate to model real complex systems.

Over the past two decades, a better paradigm – that of networks – has come to be recognized as central for the study of real-world systems. Network models are routinely applied to a wide range of questions in many research areas (Albert and Barabási 2002; Newman 2003; Boccaletti et al. 2006a, 2014). The representation of a complex system as a network, i.e., a set of discrete nodes connected in pairs by discrete links, called edges, provides a conceptual simplification that often allows researchers to gain deep analytical insights.

Since the links in a network always connect pairs of nodes, it is possible to represent any particular connection topology as a matrix, which takes the name adjacency matrix. In its simplest possible form, an adjacency matrix A is a 0–1 matrix such that Ai, j is 1 if and only if a link exists between node i and node j, and is 0 otherwise.

If no link in a network has a preferential direction, then Ai, j = Aj, i, and A is a symmetric matrix. Conversely, one may attach a directionality to the links, which is useful when modeling systems in which the interaction between two nodes has univocal origin and destination. In this case, the symmetry condition for the adjacency matrix is not necessarily fulfilled. Finally, if not all the interactions have the same intensity, then the elements of the adjacency matrix are not restricted to the domain ﹛0, 1﹜. When this happens, the network is said to be weighted, and its connectivity matrix is often called a weighted adjacency matrix.