Two-parameter nonsmooth grazing bifurcations of limit cycles: classification and open problems

This paper proposes a strategy for the classification of codimension-two grazing bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a non-generic way. Several such codimension-one events have recently been identified, causing for example period-adding or sudden onset of chaos. Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincaré map from a neighbourhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the the grazing cycle is itself degenerate (e.g. non-hyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that have discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.

The use of feed blocks as supplementation for theupland hill flock: (1) Improving organic ewe productivity and performance

This report was presented at the UK Organic Research 2002 Conference of the Colloquium of Organic Researchers (COR). Maintaining ewe performance in winter poses particular problems for organic farming in the uplands where the availability of both grazing and home produced forage may be restricted. This trial evaluated approved non-organic feed blocks as dietary supplement for ewes grazing pastures between 300 and 550 m

Energy Assessment of Pastoral Dairy Goat Husbandry from an Agroecological Economics Perspective. A Case Study in Andalusia (Spain)

This paper presents a methodological proposal of new energy sustainability indicators according to a novel accounting that follows agroecological and ecological economics criteria. Energy output is reformulated to include manure and thus consider the contribution to fertilization made by pastoral livestock farming to agroecosystems. Energy inputs calculations include the grazing resources. These new definitions and calculations allow for new formulations of the energy return on investment (EROI) as measures of the energy efficiency of livestock farming systems (final EROI and food/feed EROI). The environmental benefit of manure is estimated from the avoided energy cost of using this alternative to inorganic fertilizers (AECM). The environmental benefit of grazing is measured through the energy cost of avoiding cultivated animal feed (AECP) and its impact in terms of non-utilized agricultural area (ALCP). The comparative analysis of different livestock breeding systems in three pastoral dairy goat farms in the Sierra de Cádiz in Andalusia, southern Spain, reveals the analytical potential of the new energy sustainability indicators proposed, as well as the potential environmental benefits derived from territorial-based stockbreeding and, more specifically, grazing activities. Those benefits include gains in energy efficiency, a reduction of the dependence on non-renewable energy, and environmental costs avoided in terms of energy in extensive pastoral systems

Grazing Non-Traditional Forages

In the Heart of America region we are blessed with perennial cool-season grasses and legumes as the bulk of our forage production. Efficiently taking advantage of that perennial production is the focus of most grazing operations. Occasionally there are times when the production from our cool- season grass and legume pasture is not sufficient. In Ohio, that is late summer and late fall through winter till early spring. Nontraditional forages could be used when cool season forages are not producing. Non-traditional forages can provide a large volume, of high-quality feed during specific times of the year

Topology of Vibro-Impact Systems in the Neighborhood of Grazing

The grazing bifurcation is considered for the Newtonian model of vibro-impact systems. A brief review on the conditions, sufficient for existence of a grazing family of periodic solutions, is given. The properties of these periodic solutions are discussed. A plenty of results on the topological structure of attractors of vibro-impact systems is known. However, since the considered system is strongly nonlinear, these attractors may be invisible or, at least, very sensitive to changes of parameters of the system. On the other hand, they are observed in experiments and numerical simulations. We offer (Theorem 2) an approach which allows to explain this contradiction and give a new robust mathematical model of the non-hyperbolic dynamics in the neighborhood of grazing.Comment: Submitted to Physica