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

Abstract

Natural and engineered networks, such as interconnected neurons, ecological and social networks, coupled oscillators, wireless terminals and power loads, are characterized by an appreciable heterogeneity in the local connectivity around each node. For instance, in both elementary structures such as stars and complex graphs having scale-free topology, a minority of elements are linked to the rest of the network disproportionately strongly. While the effect of the arrangement of structural connections on the emergent synchronization pattern has been studied extensively, considerably less is known about its influence on the temporal dynamics unfolding within each node. Here, we present a comprehensive investigation across diverse simulated and experimental systems, encompassing star and complex networks of Rössler systems, coupled hysteresis-based electronic oscillators, microcircuits of leaky integrate-and-fire model neurons, and finally recordings from in-vitro cultures of spontaneously-growing neuronal networks. We systematically consider a range of dynamical measures, including the correlation dimension, nonlinear prediction error, permutation entropy, and other information-theoretical indices. The empirical evidence gathered reveals that under situations of weak synchronization, wherein rather than a collective behavior one observes significantly differentiated dynamics, denser connectivity tends to locally promote the emergence of stronger signatures of nonlinear dynamics. In deterministic systems, transition to chaos and generation of higher-dimensional signals were observed; however, when the coupling is stronger, this relationship may be lost or even inverted. In systems with a strong stochastic component, the generation of more temporally-organized activity could be induced. These observations have many potential implications across diverse fields of basic and applied science, for example, in the design of distributed sensing systems based on wireless coupled oscillators, in network identification and control, as well as in the interpretation of neuroscientific and other dynamical data.

Complex dynamical behaviors of a novel exponential hyper–chaotic system and its application in fast synchronization and color image encryption

“…The control law (20) is designed with k = 20, g = 38 to ensure the fast synchronization of the unknown hyper-chaotic systems. Figure 15 displays the complete hyperchaos synchronization of system (6) and (8). According to (37), when the controller is activated, the errors of synchronization obtained are the same as those depicted in Figure 16.…”

“…Figure 14 display the amazing hyper-chaotic attractors of the master (6) and slave (8) systems with initial conditions and parameters (7) and (9). According to Assumption 2, disturbances have been applied to the slave system (8). It is worth noting that although the systems (6) and (8) are the same, for their simulation during synchronization, we considered unequal parameters and different initial conditions.…”

“…According to Assumption 2, disturbances have been applied to the slave system (8). It is worth noting that although the systems (6) and (8) are the same, for their simulation during synchronization, we considered unequal parameters and different initial conditions. According to Figure 14, all the modes of the master-slave systems behave differently.…”

“…According to Figure 14, all the modes of the master-slave systems behave differently. Based on Assumption 1, to study chaos synchronization, the error according to systems (6) and (8), can be computed as follows: Figure 14. Hyper-chaotic time series of the master-slave systems (6) and (8).…”

“…where Z p t ð Þ À Z p t ð Þ for t.T , Z t ð Þ and Z t ð Þ are the solutions of master-slave Systems (6) and (8). Lemma 1.…”