Geometrical properties of local dynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method

We investigate the detailed dynamics of multidimensional Hamiltonian systems by studying the evolution of volume elements formed by unit deviation vectors about their orbits. The behavior of these volumes is strongly influenced by the regular or chaotic nature of the motion, the number of deviation vectors, their linear (in)dependence and the spectrum of Lyapunov exponents. The different time evolution of these volumes can be used to identify rapidly and efficiently the nature of the dynamics, leading to the introduction of quantities that clearly distinguish between chaotic behavior and quasiperiodic motion on $N$-dimensional tori. More specifically we introduce the Generalized Alignment Index of order $k$ (GALI$_k$) as the volume of a generalized parallelepiped, whose edges are $k$ initially linearly independent unit deviation vectors from the studied orbit whose magnitude is normalized to unity at every time step. The GALI$_k$ is a generalization of the Smaller Alignment Index (SALI) (GALI$_2$ $\propto$ SALI). However, GALI$_k$ provides significantly more detailed information on the local dynamics, allows for a faster and clearer distinction between order and chaos than SALI and works even in cases where the SALI method is inconclusive.Comment: 45 pages, 10 figures, accepted for publication in Physica

Economies of scale in the production of swine manure Economias de escala na produção de dejetos de suínos

Manure production on grower/finisher swine operations in the United States was examined using data from 184 grower/finisher swine operations that participated in the United States National Animal Health Monitoring System's 1995 National Swine Study. Two methods were used: one, assuming that pigs produced 8.4% of their body weight in manure each day; another using the difference between feed fed and weight gained as a proxy variable to study manure production. Using this latter approach, a production function was developed. The function exhibited diminishing returns to scale when food waste was not fed to pigs, but constant returns to scale when food waste was included in their diets. The difference between feed fed and weight gained was lower on operations that restricted entry to employees only.<br>A produção de dejetos em granjas de crescimento e terminação de suínos nos Estados Unidos foi avaliada utilizando dados de 184 granjas participantes de um estudo nacional de 1995 do "United States National Animal Health Monitoring System". Dois métodos foram usados: um considerando que suínos produzem 8,4% do seu peso corporal de dejetos por dia e o outro usando a diferença entre o alimento ingerido e o ganho de peso como um indicador para o estudo da produção de esterco. Através desse último procedimento, desenvolveu-se uma função de produção