After reviewing a number of results from geometric singular perturbation theory, we give an example of a theorem for periodic solutions in a slow manifold. This is illustrated by examples involving the van der Pol-equation and a modified logistic equation. Regarding nonhyperbolic transitions we discuss a four-dimensional relaxation oscillation and also canard-like solutions emerging from the mo...

This paper details the development and application of the RDS-SUMO-CEASIOM-EDGE rapid-CFD tool. It uses the RDS CAD model as geometry for automated meshing and CFD analysis to produce an aero-data base for control and stability analysis. It is applied to two non-conventional design proposals, an asymmetric twin-prop aircraft and an airliner with rear Open Rotor propulsion, retractable canard, a...

The Blasius problem f ′′′ + ff ′′ = 0, f(0) = −a, f ′(0) = b, f ′(+∞) = λ is investigated, in particular in the difficult and scarcely studied case b < 0 λ. The shape and the number of solutions are determined. The method is first to reduce to the Crocco equation uu′′ + s = 0 and then to use an associated autonomous planar vector field. The most useful properties of Crocco solutions appear to b...

We show that firing activity (spiking) can be regularized by noise in a FitzHugh-Nagumo (FHN) neuron model when operating slightly beyond the supercritical Hopf bifurcation (in the "canard" region). We also provide the conditions for imperfect phase locking between interspike intervals and low amplitude quasiharmonic oscillations. For the imperfect phase locking no need exists of an external si...

In this paper we use the blowup method of Dumortier and Roussarie, in the formulation due to Krupa and Szmolyan, to study the regularization of singularities of piecewise smooth dynamical systems in R. Using the regularization method of Sotomayor and Teixeira, we first demonstrate the power of our approach by considering the case of a fold line. We quickly extend a main result of Reves and Sear...

This paper is concerned with the geometry of slow manifolds of a dynamical system with two slow and one fast variable. Specifically, we study the dynamics near a folded node singularity, which is known to give rise to so-called canard solutions. Geometrically, canards are intersection curves of two-dimensional attracting and repelling slow manifolds, and they are a key element of slow-fast dyna...

§ The interactive aerodynamics of new V/STOL designs require a new approach to ground testing, where multiple properties are measured both on and away from surfaces during continuous changes in test parameters. Using a set of experiments of increasing complexity, a new capability is demonstrated for capturing surface pressure, velocity fields and vortex flow features over a range of test parame...

The work is devoted to the investigation of critical phenomena using the geometric theory of singular perturbations, namely, the black swans and canards techniques. The interest to critical phenomena is occasioned by not only of safety reason, in many cases namely the critical regime is the most effective in technological processes. The sense of criticality here is as follows. The critical regi...

We study the recently observed phenomena of torus canards. These are a higher-dimensional generalization of the classical canard orbits familiar from planar systems and arise in fast-slow systems of ordinary differential equations in which the fast subsystem contains a saddle-node bifurcation of limit cycles. Torus canards are trajectories that pass near the saddle-node and subsequently spend l...