Stability and steady state of complex cooperative systems: a diakoptic approach

Cooperative dynamics are common in ecology and population dynamics. However, their commonly high degree of complexity with a large number of coupled degrees of freedom renders them difficult to analyse. Here, we present a graph-theoretical criterion, via a diakoptic approach (divide-and-conquer) to determine a cooperative system’s stability by decomposing the system’s dependence graph into its strongly connected components (SCCs). In particular, we show that a linear cooperative system is Lyapunov stable if the SCCs of the associated dependence graph all have non-positive dominant eigenvalues, and if no SCCs which have dominant eigenvalue zero are connected by a path.

Optimizing Pinned Nodes to Maximize the Convergence Rate of Multiagent Systems with Digraph Topologies

This paper investigates how to choose pinned node set to maximize the convergence rate of multiagent systems under digraph topologies in cases of sufficiently small and large pinning strength. In the case of sufficiently small pinning strength, perturbation methods are employed to derive formulas in terms of asymptotics that indicate that the left eigenvector corresponding to eigenvalue zero of the Laplacian measures the importance of node in pinning control multiagent systems if the underlying network has a spanning tree, whereas for the network with no spanning trees, the left eigenvectors of the Laplacian matrix corresponding to eigenvalue zero can be used to select the optimal pinned node set. In the case of sufficiently large pinning strength, by the similar method, a metric based on the smallest real part of eigenvalues of the Laplacian submatrix corresponding to the unpinned nodes is used to measure the stabilizability of the pinned node set. Different algorithms that are applicable for different scenarios are develped. Several numerical simulations are given to verify theoretical results.

The k-Zero Bifurcation Theorem for m-Parameterized Systems

Given an [Formula: see text]-parameterized family of [Formula: see text]-dimensional vector fields, with an equilibrium point with linearization of eigenvalue zero with algebraic multiplicity [Formula: see text], with [Formula: see text], and geometric multiplicity one, our goal in this paper is to find sufficient conditions for the family of vector fields such that the dynamics on the [Formula: see text]-dimensional [Formula: see text]-parameterized center manifold around the equilibrium point becomes locally topologically equivalent to a given unfolding. Finally, the result is applied to the study of the Rössler system.

Ricci solitons on QR-hypersurfaces of a quaternionic space form ℚn

The purpose of this paper is to study Ricci solitons on QR-hypersurfaces M of a quaternionic space form ℚn such that the shape operator A with respect to N has one eigenvalue. We prove that Ricci soliton on QR-hypersurfaces M with eigenvalue zero is steady and for eigenvalue nonzero is shrinking.

On the nullity of a family of tripartite graphs

The eigenvalues of the adjacency matrix of a graph form the spectrum of the graph. The multiplicity of the eigenvalue zero in the spectrum of a graph is called nullity of the graph. Fan and Qian (2009) obtained the nullity set of n-vertex bipartite graphs and characterized the bipartite graphs with nullity n − 4 and the regular n-vertex bipartite graphs with nullity n − 6. In this paper, we study similar problem for a class of tripartite graphs. As observed the nullity problem in tripartite graphs does not follow as an extension to that of the nullity of bipartite graphs, this makes the study of nullity in tripartite graphs interesting. In this direction, we obtain the nullity set of a class of n-vertex tripartite graphs and characterize these tripartite graphs with nullity n − 4. We also characterize some tripartite graphs with nullity n − 6 in this class.

Further Results on the Nullity of Signed Graphs

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. A signed graph is a graph with a sign attached to each of its edges. In this paper, we apply the coefficient theorem on the characteristic polynomial of a signed graph and give two formulae on the nullity of signed graphs with cut-points. As applications of the above results, we investigate the nullity of the bicyclic signed graphΓ∞p,q,l, obtain the nullity set of unbalanced bicyclic signed graphs, and thus determine the nullity set of bicyclic signed graphs.

HUBUNGAN MATRIKS AB DAN BA PADA STRUKTUR JORDAN NILPOTEN

In this paper, we give another proof about the relationship between AB and BA with eigenvalue zero that reduced by structure Jordan for nilpoten matrix

EXPONENTIAL SYNCHRONIZATION OF NONLINEAR COUPLED DYNAMICAL NETWORKS

In this paper, we discuss exponential synchronization of nonlinear coupled dynamical networks. Sufficient conditions for both local and global exponential synchronization are given. These conditions indicate that the left and right eigenvectors corresponding to eigenvalue zero of the coupling matrix play key roles in the stability analysis of the synchronization manifold.