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The global dynamics of coupled systems of differential equations defined on an interaction network are investigated. Local dynamics at each vertex, when interactions are absent, are assumed to be simple: solutions to each vertex system are assumed to converge to an equilibrium, either on the boundary or in the interior of the feasible region. The interest is to investigate the collective behaviours of the coupled system when interactions among vertex systems are present. It was shown in Li and Shuai (2010) that, if the interaction network is strongly connected, then solutions to the coupled system synchronize at a single equilibrium. We focus on the case when the underlying network is not strongly connected and the coupled system may have mixed equilibria whose coordinates are in the interior at some vertices while on the boundary at others. We show that solutions on a strongly connected component of the network will synchronize. Considering a condensed digraph by collapsing each strongly connected component, we are able to introduce a partial order on the set P of all equilibria, and show that all solutions of the coupled system converge to a unique equilibrium P ∗ that is the maximizer in P . We further establish that behaviours of the coupled system at minimal elements of the condensed digraph determine whether the global limit P ∗ is a mixed equilibrium. The theory are applied to mathematical models from epidemiology and spatial ecology. [ABSTRACT FROM AUTHOR]

We investigate Hopf bifurcations in a delayed Nicholson's blowflies equation of neutral type, derived from the Gurtin-MacCamy model. A key parameter that determines the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Global extension of local Hopf branches is established by combining a global Hopf bifurcation theorem with a Bendixson criterion for higher dimensional ordinary differential equations. We show that a branch of slowly varying periodic solutions and a branch of fast oscillating periodic solutions coexist for all large delays. [ABSTRACT FROM AUTHOR]

Abstract: The dynamics of a neural network model in neutral form is investigated. We prove that a sequence of Hopf bifurcations occurs at the origin as the delay increases. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using normal form method and center manifold theory. Global existence of periodic solutions is established using a global Hopf bifurcation result of Krawcewicz et al. and a Bendixson''s criterion for higher dimensional ordinary differential equations due to Li and Muldowney. [Copyright &y& Elsevier]

The dynamics of a general in-host model with intracellular delay is studied. The model can describe in vivo infections of HIV-I, HCV, and HBV. It can also be considered as a model for HTLV-I infection. We derive the basic reproduction number R0 for the viral infection, and establish that the global dynamics are completely determined by the values of R0. If R0≤1, the infection-free equilibrium is globally asymptotically stable, and the virus are cleared. If R0>1, then the infection persists and the chronic-infection equilibrium is locally asymptotically stable. Furthermore, using the method of Lyapunov functional, we prove that the chronic-infection equilibrium is globally asymptotically stable when R0>1. Our results shows that for intercellular delays to generate sustained oscillations in in-host models it is necessary have a logistic mitosis term in target-cell compartments. [ABSTRACT FROM AUTHOR]

Abstract: The global-stability problem of equilibria is investigated for coupled systems of differential equations on networks. Using results from graph theory, we develop a systematic approach that allows one to construct global Lyapunov functions for large-scale coupled systems from building blocks of individual vertex systems. The approach is applied to several classes of coupled systems in engineering, ecology and epidemiology, and is shown to improve existing results. [Copyright &y& Elsevier]

This paper investigates a new method to solve the inverse problem of Rutherford backscattering (RBS) data. The inverse problem is to determine the sample structure information from measured spectra, which can be defined as a function approximation problem. We propose using radial basis function (RBF) neural networks to approximate an inverse function. Each RBS spectrum, which may contain up to 128 data points, is compressed by the principal component analysis, so that the dimensionality of input data and complexity of the network are reduced significantly. Our theoretical consideration is tested by numerical experiments with the example of the SiGe thin film sample and corresponding backscattering spectra. A comparison of the RBF method with multilayer perceptrons reveals that the former has better performance in extracting structural information from spectra. Furthermore, the proposed method can handle redundancies properly, which are caused by the constraint of output variables. This study is the first method based on RBF to deal with the inverse RBS data analysis problem. [ABSTRACT FROM AUTHOR]

Abstract: The dynamics of a Nicholson''s blowflies equation with a finite delay are investigated. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result of Wu (Trans. Amer. Math. Soc. 350 (1998) 4799), and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney (J. Differential Equations 106 (1994) 27). [Copyright &y& Elsevier]

We consider a delayed differential system that models a network of three neurons with memory. Using a global Hopf bifurcation theorem for FDE due to J. Wu and a Bendixson’s criterion for high-dimensional ODE due to Li and Muldowney, we obtain a group of sufficient conditions for the system to have multiple periodic solutions when the sum of delays is sufficiently large. [Copyright &y& Elsevier]