Abstract:Adaptation plays a fundamental role in shaping the structure of a complex network and improving its functional fitting. Even when increasing the level of synchronization in a biological system is considered as the main driving force for adaptation, there is evidence of negative effects induced by excessive synchronization. This indicates that coherence alone cannot be enough to explain all the structural features observed in many real-world networks. In this work, we propose an adaptive network model where the… Show more
“…The parameter α can be considered as a phase-lag of the interaction [46]. System (1.1)-(1.2) has attracted a lot of attention recently [26,6,7,25,35,19,39,50,45,10], since it is a first choice paradigmatic model for the modeling of the dynamics of adaptive networks. In particular, it generalizes the Kuramoto (or Kuramoto-Sakaguchi) model with fixed κ [4,28,37,49,40].…”
Dynamical systems on networks with adaptive couplings appear naturally in realworld systems such as power grid networks, social networks as well as neuronal networks. We investigate a paradigmatic system of adaptively coupled phase oscillators inspired by neuronal networks with synaptic plasticity. One important behaviour of such systems reveals splitting of the network into clusters of oscillators with the same frequencies, where different clusters correspond to different frequencies. Starting from one-cluster solutions we provide existence criteria for multicluster solutions and present their explicit form. The phases of the oscillators within one cluster can be organized in different patterns: antipodal, double antipodal, and splay type. Interestingly, multi-clusters are shown to exist where different clusters exhibit different patterns. For instance, an antipodal cluster can coexist with a splay cluster. We also provide stability conditions for one-and multi-cluster solutions. These conditions, in particular, reveal a high level of multistability.
“…The parameter α can be considered as a phase-lag of the interaction [46]. System (1.1)-(1.2) has attracted a lot of attention recently [26,6,7,25,35,19,39,50,45,10], since it is a first choice paradigmatic model for the modeling of the dynamics of adaptive networks. In particular, it generalizes the Kuramoto (or Kuramoto-Sakaguchi) model with fixed κ [4,28,37,49,40].…”
Dynamical systems on networks with adaptive couplings appear naturally in realworld systems such as power grid networks, social networks as well as neuronal networks. We investigate a paradigmatic system of adaptively coupled phase oscillators inspired by neuronal networks with synaptic plasticity. One important behaviour of such systems reveals splitting of the network into clusters of oscillators with the same frequencies, where different clusters correspond to different frequencies. Starting from one-cluster solutions we provide existence criteria for multicluster solutions and present their explicit form. The phases of the oscillators within one cluster can be organized in different patterns: antipodal, double antipodal, and splay type. Interestingly, multi-clusters are shown to exist where different clusters exhibit different patterns. For instance, an antipodal cluster can coexist with a splay cluster. We also provide stability conditions for one-and multi-cluster solutions. These conditions, in particular, reveal a high level of multistability.
“…More specifically, we study rings of nonlocally coupled oscillators based on the Kuramoto-Sakaguchi model [26,27] with an additional adaptation dynamics of the coupling weights. For an all-to-all coupling topology similar models have been recently a e-mail: rico.berner@physik.tu-berlin.de studied [28][29][30][31][32][33][34][35][36][37], but very little is known about the dynamics of these systems if the base topology is more complex [38]. On globally coupled networks, adaptive Kuramoto-Sakaguchi type models have been shown to exhibit diverse complex dynamical behavior.…”
In this article, we analyze a nonlocal ring network of adaptively coupled phase oscillators. We observe a variety of frequency-synchronized states such as phase-locked, multicluster and solitary states. For an important subclass of the phase-locked solutions, the rotating waves, we provide a rigorous stability analysis. This analysis shows a strong dependence of their stability on the coupling structure and the wavenumber which is a remarkable difference to an all-to-all coupled network. Despite the fact that solitary states have been observed in a plethora of dynamical systems, the mechanisms behind their emergence were largely unaddressed in the literature. Here, we show how solitary states emerge due to the adaptive feature of the network and classify several bifurcation scenarios in which these states are created and stabilized.
“…Consequently, a predator species in food webs has to strengthen the weight of links to other prey species and thus compensate for the weight lost (figure 1 b ). A few studies have proposed mechanics to adjust link weights [17,20,21], but most are grounded on structure-based approaches, without considering the dynamics of the systems. Figure 1Structural adaptation and dynamical adaptation after species loss.…”
Mutualistic networks, which describe the ecological interactions between multiple types of species such as plants and pollinators, play a paramount role in the generation of Earth’s biodiversity. The resilience of a mutualistic network denotes its ability to retain basic functionality when errors and failures threaten the persistence of the community. Under the disturbances of mass extinctions and human-induced disasters, it is crucial to understand how mutualistic networks respond to changes, which enables the system to increase resilience and tolerate further damages. Despite recent advances in the modelling of the structure-based adaptation, we lack mathematical and computational models to describe and capture the co-adaptation between the structure and dynamics of mutualistic networks. In this paper, we incorporate dynamic features into the adaptation of structure and propose a co-adaptation model that drastically enhances the resilience of non-adaptive and structure-based adaptation models. Surprisingly, the reason for the enhancement is that the co-adaptation mechanism simultaneously increases the heterogeneity of the mutualistic network significantly without changing its connectance. Owing to the broad applications of mutualistic networks, our findings offer new ways to design mechanisms that enhance the resilience of many other systems, such as smart infrastructures and social–economical systems.
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