Motivated by pulse-replication phenomena observed in the FitzHugh--Nagumo equation, we investigate traveling pulses whose slow/fast profiles exhibit canard-like transitions. We show that spectra of PDE linearization about such may contain many point eigenvalues accumulate onto a union curves as slow scale parameter approaches zero. The limit sets are related to absolute spectrum homogeneous res...

We introduce a new class of excitable systems with two-dimensional fast dynamics that includes inertia. A novel transition from excitability to relaxation oscillations is discovered where the usual Hopf bifurcation is followed by a cascade of period doubled and chaotic small excitable attractors and, as they grow, by a new type of canard explosion where a small chaotic background erratically bu...

This paper provides algorithms in order to solve an interval implicit function of the Poincaré map generated by a continuous piecewise linear (CPWL) vector field, with the use of interval arithmetic. The algorithms are implemented with the use of MAT LAB and INT LAB. We present an application to verification of canards in two-dimensional CPWL vector field appearing in nonlinear piecewise linear...

The geometric approach to singular perturbation problems is based on powerful methods from dynamical systems theory. These techniques have been very successful in the case of normally hyperbolic critical manifolds. However, at points where normal hyperbolicity fails, the well-developed geometric theory does not apply. We present a method based on blow-up techniques, which leads to a rigorous ge...

A popular canard among critics of psychiatry is that psychiatric disorders are never listed in pathology textbooks. This erroneous claim is sometimes used to argue that some diseases such as schizophrenia are not "real" diseases, but merely metaphorical inventions of psychiatrists. In reality, many pathology and pathophysiology texts now recognize schizophrenia as bona fide disease, and physici...

We present a geometric singular perturbation analysis of a chemical oscillator. Although the studied three-dimensional model is rather simple its dynamics are quite complex. In the original scaling the problem has a folded critical manifold which additionally becomes tangent to the fast bers in a region relevant to the dynamics. Thus normal hyperbolicity of the critical manifold is lost in two ...

Slow-fast systems on the two-torus T 2 provide new effects not observed for systems on the plane. Namely, there exist families without auxiliary parameters that have attracting canard cycles for arbitrary small values of the time scaling parameter ε. In order to demonstrate the new effect, we have chosen a particularly simple family, namely ẋ = a − cosx − cos y, ẏ = ε, a ∈ (1, 2) being fixed. T...