On a special type of border-collision bifurcations occurring at infinity

In piecewise-smooth dynamical systems, the regions of existence of a periodic orbit are typically parameter sub-spaces confined by border-collision bifurcations of this orbit. We demonstrate that additionally to the usual border-collision bifurcations occurring at finite points in the state space there exist also border-collision bifurcations occurring at infinity.     

Comments on some fixed point theorems in hyperconvex metric spaces

In this paper, we show that a number of known fixed point theorems for the Fan-Browder type maps or acyclic maps defined on (subsets of) hyperconvex metric spaces are simple consequences of the previously known theorems for corresponding maps defined on generalized convex spaces.     

On the purposeful coarsening of smooth vector fields

This paper argues for the possibility of purposely approximating smooth vector fields with highly localized variability in terms of piecewise smooth vector fields for the purpose of analyzing the bifurcation characteristics of the corresponding dynamical systems. Here, emphasis is placed on the changes in system response that result as a periodic trajectory begins to incorporate a brief flow segment in the region of high variability under variations in some system parameter. In particular, it is shown that tools from the theory of grazing bifurcations in piecewise-smooth systems may be employed to qualitatively predict the bifurcation scenario associated with such a transition both in terms of the shape of the branch of periodic trajectories and in terms of the persistence of a local attractor in the vicinity of the original periodic trajectory.     

Experimental and numerical verification of bifurcations and chaos in cam-follower impacting systems

In this paper, we present the design, modelling and experimental validation of a novel experimental cam-follower rig for the analysis of bifurcations and chaos in piecewise-smooth dynamical systems with impacts. Experimental results are presented for a cam-follower system characterized by a radial cam and a flat-faced follower. Under variation of the cam rotational speed, the follower is observed to detach from the cam and then show the emergence of periodic impacting behaviour characterized by many impacts and chattering. Further variations of the cam speed cause the sudden transition to seemingly aperiodic behaviour. These results are compared with the numerical simulation of a mathematical model of the system which shows the same qualitative behaviour. Excellent quantitative agreement is found between the numerical and experimental results.     

Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems

A rich variety of dynamical scenarios has been shown to occur when a fixed point of a non-smooth map undergoes a border-collision. This paper concerns a closely related class of discontinuity-induced bifurcations, those involving equilibria of n-dimensional piecewise-smooth flows. Specifically, transitions are studied which occur when a boundary equilibrium, that is one lying within a discontinuity manifold, is perturbed. It is shown that such equilibria can either persist under parameter variations or can disappear giving rise to different bifurcation scenarios. Conditions to classify among the possible simplest scenarios are given for piecewise-smooth continuous, Filippov and impacting systems. Also, we investigate the possible birth of other attractors (e.g. limit cycles) at a boundary-equilibrium bifurcation.     

Miridae control using sex-pheromone traps. Modeling,analysis and simulations

Cocoa mirid, Sahlbergella singularis, is known to be one of the major pests of cocoa in West Africa. In this paper, we consider a biological control method, based on mating disrupting, using artificial sex pheromones, and trapping, to limit the impact of mirids in plots. We develop and study a piece-wise smooth delayed dynamical system. Based on previous results, a theoretical analysis is provided in order to derive all possible dynamics of the system. We show that two main threshold parameters exist that will be useful to derive long term successful control strategies for different level of infestation. We illustrate and discuss our results when cacao pods production is either constant along the year or seasonal. To conclude, we provide future perspectives based on this work.