Abstract
In this contribution, we present the synchronization in dynamical complex networks with varying couplings. We identify two kinds of variations—(i) Non autonomous (Time-varying) couplings: where the coupling strength depends exclusively on time, (ii) Autonomous or Varying couplings (evolution) where the coupling strength depends on the behavior of the interconnected systems. The coupling strength in (i) is exogenous whereas in (ii) the coupling strength is endogenous and is defined by the states of the systems in the nodes. The exponential stability of the synchronization is ensured for the non autonomous couplings, due to the imposition of the coupling strength. Whereas, in the case of evolutionary couplings the exponential stability of the synchronization is not guaranteed for all time, due to the couplings are not controlled or imposed. We present an overview of these features in complex networks and illustrated by means of numerical examples.
Highlights
A complex network is a set of coupled elements with features that do not occur in simple networks such as lattices or random graphs, but often occur when trying to model real-world systems.Many human-made and natural systems are described by models of complex networks, such as the citation network of scientific papers, electrical networks, wireless communication networks, food chains, social networks, ecosystems, and so on [1,2,3,4,5,6].The concept of complex networks introduced by Watts and Strogatz (WS) [7] aims to describe the transition from a regular lattice to a random graph
The synchronization with evolutionary couplings is achieved, the synchronization is not exponentially stable for all time, this is the condition λ2 (t) < d is not satisfied for all t > 0 as it is shown in Figure 9, where the red line dives the exponential stability from the stability of the synchronization state, this is due to the parameters which define the value d
Note that for non autonomous exogenous couplings, the synchronization is controlled by the selection of the couplings, in other words it is always possible to fix the value of the couplings in order to satisfy the condition λ2 (t) < d for all t > 0 whereas in case of evolutionary couplings, the synchronization is reached by the collectivity of the nodes and as the couplings are not controlled or imposed the synchronization can or cannot be exponentially
Summary
A complex network is a set of coupled elements with features that do not occur in simple networks such as lattices or random graphs, but often occur when trying to model real-world systems. Changes along time refer to changes in the couplings between nodes which are defined as function purely of time In this sense, the changes in the network are dictated by an external agent, and the nodes in the networks do not influence their couplings, for instance in Reference [17] the authors consider varying couplings but a control action is implemented to achieve synchronization in a defined reference. The changes in the network are dictated by an external agent, and the nodes in the networks do not influence their couplings, for instance in Reference [17] the authors consider varying couplings but a control action is implemented to achieve synchronization in a defined reference Another example is in brain deceases or disorders such as Parkinson, Alzheimer’s, and so forth, where external deep brain stimulation is provided to modify the behavior of certain parts of the brain [18,19,20]. The paper is organized as follows, the model for a dynamical network is presented, in Section 3, models for networks with chaotic couplings and evolving couplings are presented, alongside numerical examples to corroborate our results, and in Section 4 some conclusions are provided
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