Discontinuities are a common feature of physical models in engineering and biological systems, e.g. stick-slip due to friction, electrical relays or gene regulatory networks. The computation of basins of attraction of such nonsmooth systems is challenging and requires special treatments, especially regarding numerical integration. In this paper, we present a numerical routine for computing basins of attraction (BA) in nonsmooth systems with sliding, (so-called Filippov systems). In particular, we extend the Simple Cell Mapping (SCM) method to cope with the presence of sliding solutions by exploiting an event-driven numerical integration routine specifically written for Filippov systems. Our algorithm encompasses a method for dynamic construction of the cell state space so that a lower number of integration steps are required. Moreover, we incorporate an adaptive strategy of the simulation time to render more efficiently the computation of basins of attraction. We illustrate the effectiveness of our algorithm by computing basins of attraction of a sliding control problem and a dry-friction oscillator.

Background : Complete esophago-gastric necrosis after caustic ingestion is a challenging surgical scenario for reconstruction of the upper digestive transit. Aim : To present a surgical technique for reconstruction of the upper digestive tract after total esophagectomy and gastrectomy due to esophageal and gastric necrosis Method: The transit was re-established by means of a pharyngo-ileo-colic interposition with microsurgical arterial and venous anastomosis for augmentation of blood supply. Colo-duodeno-anastomosis and ileo-transverse colic anastomosis were performed for complete digestive transit reconstruction. Result: This procedure was applied in a case of 41 years male attempted suicide by ingesting alkali caustic liquid (concentrated sodium hydroxide). Total necrosis of the esophagus and stomach occurred, which required initially total esophago-gastrectomy, closure at the level of the crico-pharyngeal sphincter and jejunostomy for enteral feeding with a highly deteriorated quality of life. The procedure was performed later and there were no major early and late postoperative complications and normal nutritional conditions were re-stablished. Conclusion: The procedure is feasible and must be managed by multidisciplinary team in order to re-establish a normal quality of life.

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The concept of Sustainable Development has given rise to multiple interpretations. In this article, it is proposed that Sustainable Development should be interpreted as the capacity of territory, community, or landscape to conserve the notion of well-being that its population has agreed upon. To see the implications of this interpretation, a Brander and Taylor model, to evaluate the implications that extractivist policies have over an isolated community and cooperating communities, is proposed. For an isolated community and through a bifurcation analysis in which the Hopf bifurcation and the heteroclinic cycle bifurcation are detected, 4 prospective scenarios are found, but only one is sustainable under different extraction policies. In the case of cooperation, the exchange between communities is considered by coupling two models such as the one defined for the isolated community, with the condition that their transfers of renewable resources involve conservation policies. Since human decisions do not occur in a continuum, but rather through jumps, the mathematical model of cooperation used is a Filippov System, in which the dynamics could involve two switching manifolds of codimension one and one switching manifold of codimension two. The exchange in the cooperation model, for specific parameter arrangements, exhibits n -periodic orbits and chaos. It is notable that, in the cases in which the system shows sliding, it could be interpreted as a recovery delay related to the time needed by the deficit community to recover, until its dependence on the other community stops. It is concluded (1) that a sustainability analysis depends on the way well-being is defined because every definition of well-being is not necessarily sustainable, (2) that sustainability can be visualized as invariant sets in the nonzero region of the space of states (equilibrium points, n -periodic orbits, and strange attractors), and (3) that exchange is key to the prevalence of the human being in time. The results question us on whether Sustainable Development is only to keep us alive or if it also implies doing it with dignity.