On the emergence of chaos in dynamical networks

Abstract

We investigate how changes of specific topological features result on transitions among different bounded behaviours in dynamical networks. In particular, we focus on networks with identical dynamical systems, synchronised to a common equilibrium point, then a transition into chaotic behaviour is observed as the number of nodes and the strength of their coupling changes. We analyse the network's transverse Lyapunov exponents (tLes) to derive conditions for the emergence of bounded complex behaviour on different basic network models. We find that, for networks with a given number of nodes, chaotic behaviour emerges when the coupling strength is within a specific bounded interval; this interval is reduced as the number of nodes increases. Furthermore, the endpoints the emergence interval depend on the coupling structure of network. We also find that networks with homogeneous connectivity, such as regular lattices and small-world networks are more conducive to the emergence of chaos than networks with heterogeneous connectivity like scale-free and star-connected graphs. Our results are illustrated with numerical simulations of the chaotic benchmark Lorenz systems, and to underline their potential applicability to real-world systems, our results are used to establish conditions for the chaotic activation of a network of electrically coupled pancreatic -cell models.

Connectivity Influences on Nonlinear Dynamics in Weakly-Synchronized Networks: Insights From Rössler Systems, Electronic Chaotic Oscillators, Model and Biological Neurons

“…The specific trend, however, seems to depend on a multitude of features, including the topology of the entire network. As such, our results corroborate and extend recent work in this direction, also cautioning against assuming that a simple relationship holds universally [35], [62], [63]. Interestingly, the effect of connectivity was, at least on the surface, effectively diametrically opposite in scenarios of strong and weak determinism of the elemental dynamics.…”

“…A limited number of previous studies, albeit considering a different coupling mechanism, had addressed the effect of connectivity on the emergence of a collective chaotic state, including the need for a coupling strength within a bounded interval [62], [63]; compared to those earlier results, here we demonstrated in a rather more explicit manner the effects of degree and coupling strength, by means of investigating in detail the paradigmatic case of star networks. Because the coupled systems have, in principle, sufficient dimensionality to support hyperchaos, future work should explicitly derive the spectrum of Lyapunov exponents, to ascertain whether the observed higher-dimensional dynamics are associated with an increasing number of positive exponents; such an undertaking is beyond the scope of this work.…”

Global history, the emergence of chaos and inducing sustainability in networks of socio-ecological systems

“…First, an increase in dissipation within an ensemble characterized by a fixed coupling force and a set number of elements can initiate the onset of chaotic behavior [45,46]. Additionally, modifications of network structure have the potential to engender chaos or even hyperchaos, exemplifying the sensitivity of these systems to alterations in their composition [25,[45][46][47][48][49][50][51].…”

“…Also, the coupling strength between the elements plays a pivotal role in the manifestation of chaos, exhibiting a highly specific and nuanced influence on the overall dynamical behavior [46,54]. Lastly, the heterogeneity of the components seems to play a role in the way chaotic dynamics arise [47,55,56]. Moreover, in ecological networks, chaotic behaviors are found for a wide set of dynamical processes, including percolation [57].…”