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• Complete analysis of generalized Gause models with non-analytical functional responses. • Existence of family of closed orbits in case of functional response of Holling Type I. • Existence of family of closed orbits in case of a concave differentiable non-analytical functional response. • Uniqueness of limit cycle in case of a concave differentiable non-analytical functional response. • Co-existence of a stable equilibrium with a stable limit cycle in case of a sigmoidal differentiable non-analytical functional response. In this paper we study the generalized Gause model, with a logistic growth rate for the prey in absence of the predator, a constant death rate for the predator and for several different classes of functional response, all non-analytical. First we consider the piecewise-linear functional response of Holling type I, which essentially has a linear functional response on a bounded interval and a constant functional response for large enough prey density. Next we consider differentiable modifications of this type of functional response, one being a concave down function, the other one being a sigmoidal function. Our main interest is the number of closed orbits of the systems under consideration and the global stability of the system. We compare the generalized Gause model with a functional response that is non-analytical with the generalized Gause model with a functional response that is analytical (e.g., Holling type II or III) and show that the behaviour in the first case is more complicated. As examples of this more complicated behaviour we mention: the co-existence of a stable equilibrium with a stable limit cycle and the existence of a family of closed orbits. [ABSTRACT FROM AUTHOR]

Power grids vulnerability is a key issue in society. A component failure may trigger cascades of failures across the grid and lead to a large blackout. Within complex network analysis, structural vulnerabilities of power grids have been studied mostly using purely topological approaches, which assumes that flow of power is dictated by shortest paths. However, this fails to capture the real flow characteristics of power grids. We have proposed a flow redistribution mechanism that closely mimics the flow in power grids using the power transfer distribution factor (PTDF). We apply the model to the European high-voltage grid to carry out a comparative study for a number of centrality measures. ‘Centrality’ gives an indication of the criticality of network components. We also show a brief comparison of the end results with other power grid systems to generalise the result.

In this paper we prove that if a quadratic system with four finite singularities contains a weak singularity, then there exists at most one limit cycle not surrounding this singularity. We also show that if the limit cycles of quadratic systems with four finite singularities are distributed over two nests then each nest contains exactly one limit cycle.

5proposed a theorem to guarantee the uniqueness of limit cycles for the generalized Liénard systemdx/dt=h(y)−F(x),dy/dt=−g(x). We will give a counterexample to their theorem. It will be shown that their theorem is valid only ifF(x) is monotone on certain intervals. For this case we give a shorter proof and we also show that the limit cycle is hyperbolic. If the condition on the monotonicity ofF(x) is violated then we need an additional condition to guarantee the uniqueness of the limit cycle. Examples are provided to illustrate our results.

Abstract.:  The aim of this paper is to prove the uniqueness of isolated periodic solutions (i.e. limit cycles) in two simple models for microparasitic and macroparasitic diseases. Both models are described by systems of planar autonomous ordinary differential equations. After transformation of these systems to generalized Linard systems, we will apply a modified theorem of Zhang and Dulac’s criterion to prove the uniqueness of limit cycles.