Ascertaining when a basin is Wada: the merging method

Trying to imagine three regions separated by a unique boundary seems a difficult task. However, this is exactly what happens in many dynamical systems showing Wada basins. Here, we present a new perspective on the Wada property: A Wada boundary is the only one that remains unaltered under the action of merging the basins. This observation allows to develop a new method to test the Wada property, which is much faster than the previous ones. Furthermore, another major advantage of the merging method is that a detailed knowledge of the dynamical system is not required

The saddle-straddle method to test for Wada basins

First conceived as a topological construction, Wada basins abound in dynamical systems. Basins of attraction showing the Wada property possess the particular feature that any small perturbation of an initial condition lying on the boundary can lead the system to any of its possible outcomes. The saddle-straddle method, described here, is a new method to identify the Wada property in a dynamical system based on the computation of its chaotic saddle in the fractalized phase space. It consists of finding the chaotic saddle embedded in the boundary between the basin of one attractor and the remaining basins of attraction by using the saddle-straddle algorithm. The simple observation that the chaotic saddle is the same for all the combinations of basins is sufficient to prove that the boundary has the Wada property

Wada property in systems with delay

Delay differential equations take into account the transmission time of the information. These delayed signals may turn a predictable system into chaotic, with the usual fractalization of the phase space. In this work, we study the connection between delay and unpredictability, in particular we focus on the Wada property in systems with delay. This topological property gives rise to dramatical changes in the final state for small changes in the history functions

Reducing the number of time delays in coupled dynamical systems

When several dynamical systems interact, the transmission of the information between them necessarily implies a time delay. When the time delay is not negligible, the study of the dynamics of these interactions deserve a special treatment. We will show here that under certain assumptions, it is possible to set to zero a significant amount of time-delayed connections without altering the global dynamics. We will focus on graphs of interactions with identical time delays and bidirectional connections. With these premises, it is possible to find a configuration where a number $n_z$ of time delays have been removed with $n_v-1 \leq n_z \leq n_v^2/4$, where $n_v$ is the number of dynamical systems on a connected graph

Chaotic dynamics and fractal structures in experiments with cold atoms

We use tools from nonlinear dynamics to the detailed analysis of cold atom experiments. A powerful example is provided by the recent concept of basin entropy which allows to quantify the final state unpredictability that results from the complexity of the phase space geometry. We show here that this enables one to reliably infer the presence of fractal structures in phase space from direct measurements. We illustrate the method with numerical simulations in an experimental configuration made of two crossing laser guides that can be used as a matter wave splitter

Meat consumption after disaggregation of meat dishes in a cohort of British adults in 1989 and 1999 in relation to diet quality

Objectives—The aim of the study was to quantify more precisely the meat intake of a cohort of adults in the UK by disaggregating composite meat dishes. Subjects/Methods—Subjects were members of the MRC National Survey of Health and Development, 1946 birth cohort. Five-day diaries were collected from 2256 men and women in 1989 and 1772 men and women in 1999. From the details provided, composite meat dishes were broken down into their constituent parts and the meat fraction added to meat only portions. Meat intake was classified as red meat, processed meat and poultry. Results—Meat consumption without disaggregation of meat dishes resulted in a mean overestimation of 50% in men and 33% in women. Red meat consumption fell between 1989 and 1999 from 51.7 to 41.5g/day in men and 35.7 to 30.1g/day in women. Poultry consumption rose from 21.6 to 32.2g./day in men and 18.2 to 29.4 g/day in women. Re-calculating red meat intakes resulted in the percentage of subjects in 1999 consuming more the recommendation of the World Cancer Research Fund falling from 30% to 12 %. Increasing consumption of red and processed meat was associated with increased intakes of energy, fat, haem iron, zinc and vitamin B12 and lower intake of fibre. Increased sodium intake was associated with increased consumption of processed meat. Conclusions—Disaggregation of meat dishes provided a more precise estimate of meat consumption. The quantity of red or processed meat in the diet was reflected in the nutrient content of the entire diet

Understanding communication pathways to foster community engagement for health improvement in North West Pakistan

Background: This paper describes the community engagement process undertaken to ascertain the focus, development and implementation of an intervention to improve iodised salt consumption in rural communities in North West Pakistan. The Jirga is a traditional informal structure, which gathers men respected within their community and acts in a governing and decision making capacity in the Pukhtoon culture. The Jirga system had a dual purpose for the study; to access men from the community to discuss the importance of iodised salt, and as an engagement process for the intervention. Methods: A number of qualitative data collection activities were undertaken, with Jirga members and their wives, male and female outreach workers and two groups of women, under and over forty years old. The aim of these were to highlight the communication channels and levers of influence on health behaviour, which were multiple and complex and all needed to be taken into consideration in order to ensure successful and locally sensitive community engagement. Results: Communication channels are described within local families and the communities around them. The key influential role of the Jirga is highlighted as linked both to the standing of its members and the community cohesion ethos that it embodies. Engaging Jirga members in discussions about iodised salt was key in designing an intervention that would activate the most influential levers to decision making in the community. Gendered decision making-processes within the household have been highlighted as restricting women’s autonomy. Whilst in one respect our data confirm this, a more complex hierarchy of decisional power has been highlighted, whereby the concept of ‘wisdom’, an amalgamation of age, experience and education, presents important possibilities. Community members with the least autonomy are the youngest uneducated females, who rely on a web of socially and culturally determined ways to influence decision-making. Conclusions: The major lines of communication and influence in the local community described are placed within the wider literature on community engagement in health improvement. The process of maximisation of local cultural knowledge as part of a community engagement effort is one that has application well beyond the particular setting of this study