José Pedro Gaivão

Chaos in the square billiard with a modified reflection law

The purpose of this paper is to study the dynamics of a square billiard with a non-standard reflection law such that the angle of reflection of the particle is a linear contraction of the angle of incidence. We present numerical and analytical arguments that the nonwandering set of this billiard decomposes into three invariant sets, a parabolic attractor, a chaotic attractor, and a set consisting of several horseshoes. This scenario implies the positivity of the topological entropy of the billiard, a property that is in sharp contrast with the integrability of the square billiard with the standard reflection law.

Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift-Hohenberg equation as an example

We study homoclinic orbits of the Swift-Hohenberg equation near a Hamiltonian-Hopf bifurcation. It is well known that in this case the normal form of the equation is integrable at all orders. Therefore the difference between the stable and unstable manifolds is exponentially small and the study requires a method capable of detecting phenomena beyond all algebraic orders provided by the normal form theory. We propose an asymptotic expansion for a homoclinic invariant which quantitatively describes the transversality of the invariant manifolds. We perform high-precision numerical experiments to support the validity of the asymptotic expansion and evaluate a Stokes constant numerically using two independent methods.

Dissipative outer billiards: a case study

We study dissipative polygonal outer billiards, i.e. outer billiards about convex polygons with a contractive reflection law. We prove that dissipative outer billiards about any triangle and the square are asymptotically periodic, i.e. they have finitely many global attracting periodic orbits. A complete description of the bifurcations of the periodic orbits as the contraction rates vary is given. For the square billiard, we also show that the asymptotic periodic behaviour is robust under small perturbations of the vertices and the contraction rates. Finally, we describe some numerical experiments suggesting that dissipative outer billiards about regular polygon are generically asymptotically periodic.

ERGODICITY OF POLYGONAL SLAP MAPS

Polygonal slap maps are piecewise affine expanding maps of the interval obtained by projecting the sides of a polygon along their normals onto the perimeter of the polygon. These maps arise in the study of polygonal billiards with non-specular reflection laws. We study the absolutely continuous invariant probabilities (acips) of the slap maps for several polygons, including regular polygons and triangles. We also present a general method for constructing polygons with slap maps with more than one ergodic acip.

Observer design in switching control of neuromuscular blockade: clinical cases

This paper concerns the application of multiple model switched methods to the control of neuromuscular blockade of patients undergoing anaesthesia. Since the model representing the neuromuscular blockade process is subject to a high level of uncertainty due both to inter-patient variability and time variations, switched methods provide the adaptation capability needed to achieve the desired performance. The paper contributions are twofold: first, it is shown that, for the type of process control problem considered, the design of the associated observer must be carefully performed. Guidelines are provided for adequate selection of the characteristic polynomial defining the observer error dynamics. Second, clinical results using atracurium as blocking agent are reported in order to illustrate the use of the proposed control structure in actual clinical practice

Polygonal Billiards with Strongly Contractive Reflection Laws: A Review of Some Hyperbolic Properties

We provide an overview of recent results concerning the dynamics of polygonal billiards with strongly contractive reflection laws.