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This paper studies a heterogeneous multiplex network model that allows different dynamics in different layers. We explore intralayer synchronization of the multiplex network under distinct types of interlayer connections. From the perspective of spectral graph theory, we propose a set of edge weight requirements to synchronize the multiplex network. Focusing on the effect of interlayer connections to intralayer synchronization, it is found that a multiplex network can achieve intralayer synchronization with a large enough interlayer coupling strength even if a single network of one layer cannot synchronize by itself. In fact, the synchronizability of the multiplex network is found to be stronger than that of the single-layer network. These results provide insights into the practical application of multiplex network theory in engineering networks.

This special issue of the IEEE Journal on Emerging and Selected Topics in Circuits and Systems (JETCAS) aims at identifying, addressing, and disseminating state-of-the-art research in theory of complex systems with applications to smart grids.

This special issue of the IEEE Journal on Emerging and Selected Topics in Circuits and Systems (JETCAS) aims at identifying, addressing, and disseminating state-of-the-art research in theory of complex systems with applications to smart grids.

This paper proposes lower bounds for the coupling strengths of oscillators in directed networks to guarantee global synchronization. The novel idea of graph comparison from spectral graph theory is employed so that the combinatorial features of a given network can be fully utilized to simplify computations. For large networks that can be decomposed into a set of smaller strongly connected components, the comparison can be carried out at the local level as well. To validate theoretical analysis, examples are provided to demonstrate how to apply the proposed methodologies to typical directed complex networks. [ABSTRACT FROM AUTHOR]

AbstractIt is well known that any integer khas a multiple consisting of only the digits 1 and 0. As an extension of this result, we study integers of the form 111 ⋯ 000 or 111 ⋯ 111 that are a multiple of k. We show that if k> 2 and kis not a power of 3, then the multiple can be chosen to have at most k− 1 digits.

Using spectral graph theory and especially its graph comparison techniques, we propose new methodologies to allocate coupling strengths to guarantee global complete synchronization in complex networks. The key step is that all the eigenvalues of the Laplacian matrix associated with a given network can be estimated by utilizing flexibly topological features of the network. The proposed methodologies enable the construction of different coupling-strength combinations in response to different knowledge about subnetworks. Adaptive allocation strategies can be carried out as well using only local network topological information. Besides formal analysis, we use simulation examples to demonstrate how to apply the methodologies to typical complex networks. [ABSTRACT FROM AUTHOR]

AbstractThe Cauchy–Binet theorem is the main step in proving the multiplicative property of compound matrices. We present here a novel elementary proof of a generalization of the Cauchy–Binet theorem using properties of square matrices.

Abstract: A classical result of Wolfowitz states that an inhomogeneous Markov chain is weakly ergodic if the transition matrices are drawn from a finite set of indecomposable and aperiodic matrices and the products of transition matrices are also indecomposable and aperiodic. Since products of indecomposable and aperiodic matrices can be decomposable, any finite set of indecomposable and aperiodic transition matrices does not guarantee weak ergodicity. We present conditions for weak ergodicity which are simpler to verify and are related to properties of the graph of the transition matrices. [Copyright &y& Elsevier]

We study synchronization in an array of coupled identical nonlinear dynamical systems where the coupling topology is expressed as a directed graph and give synchronization criteria related to properties of a generalized Laplacian matrix of the directed graph. In particular, we extend recent results by showing that the array synchronizes for sufficiently large cooperative coupling if the underlying graph contains a spanning directed tree. This is an intuitive yet nontrivial result that can be paraphrased as follows: if there exists a dynamical system which influences directly or indirectly all other systems, then synchronization is possible for strong enough coupling. The converse is also true in general.

Chua's oscillator is the only real physical object known to date in which chaotic behavior has been observed experimentally and numerically, and proved rigorously. In summarizing the chaotic phenomena observed so far from the oscillator we emphasize its universality by showing how the dynamical phenomena from other 3D oscillators can be reproduced by using Chua's oscillator. The possibility of its use as an elementary cell in cellular neural networks is briefly discussed.

In this paper we present more analytical results about the complex behavior of a second order digital filter with overflow nonlinearity. We explore the parameter space to obtain a taxonomy of the different behaviors that occurs. In particular, we give a complete description of the chaotic behavior of the map F (which models the second order digital filter) in the parameter space (a,b). We prove that in the region R~/sub 5/ (the closure of R/sub 5/, where R/sub 5/={(a,b):b<-a+1, b<a+1, b>-1}) F is not chaotic; in the region |b|<1 and (a,b)∉R~/sub 5/, F has a generalized hyperbolic attractor; and in the region |b|>1, if (a,b) are integers and b=-2(a-1), then F is an exact map. In addition, we obtain some results concerning the fractal behavior of the map F. We find an estimate of the Hausdorff dimension of the generalized hyperbolic attractor. We obtain results regarding the symbolic dynamics of F. For example, we prove that the set of points with aperiodic admissible sequences in the case |a|<2 and b=-1 is uncountable.

We present image data hiding and authentication schemes based on halftoning and coordinate projection. The proposed data hiding scheme can embed images of the same size and similar bit depth into the cover image and robustness against compression is demonstrated. The image authentication scheme is based on the data hiding scheme and can detect, localize, and repair the tampered area of the image. Furthermore, the self-repairing feature of the authentication scheme has a hologram-like quality; any portion of the image can be used to reconstruct the entire image, with a greater quality of reconstruction as the portion size increases.