A study of a three-dimensional competitive Lotka–Volterra system

In this paper we will consider a community of three mutually competing species modeled by the Lotka–Volterra system: $$ {\left\{ {\dot x} \right._i} = {x_i}\left( {{b_i} - \sum\limits_{i = 1}^3 {{a_{ij}}{x_j}} } \right),i = 1,2,3 $$ where xi(t) is the population size of the i-th species at time t, Ẋi denote $${{dxi} \over {dt}}$$ and aij, bi are all strictly positive real numbers. This system of ordinary differential equations represent a class of Kolmogorov systems. This kind of systems is widely used in the mathematical models for the dynamics of population, like predator-prey models or different models for the spread of diseases. A qualitative analysis of this Lotka-Volterra system based on dynamical systems theory will be performed, by studying the local behavior in equilibrium points and obtaining local dynamics properties.

Hybrid frequency domain control for large flexible structures

A hybrid finite frequency controller is proposed for the vibration suppression of a large flexible structure mounted with collocated sensors and actuators. The controller has passive characteristics at low frequencies and small gain characteristics at high frequencies. Compared with a strictly positive real controller based on the standard Kalman–Yakubovich–Popov lemma, the hybrid finite frequency controller has less energy consumption but can obtain approximately identical performance. Furthermore, when the plant passivity is violated at high frequencies by noncollocation of sensors and actuators, the strictly positive real controller based on the Kalman–Yakubovich–Popov lemma is no longer able to attenuate the vibration of the large flexible structures, while the hybrid finite frequency controller is effective in suppressing the vibration and avoiding spillover instability. Simulation results are presented to validate the effectiveness of the hybrid finite frequency controller.