Canard Trajectories in 3D piecewise linear systems

Canard Explosion and Relaxation Oscillation in Planar, Piecewise-Smooth, Continuous Systems

Classical canard explosion results in smooth systems require the vector field to be at least C, since canard cycles are created as the result of a Hopf bifurcation. The work on canards in nonsmooth, planar systems is recent and has thus far been restricted to piecewise-linear or piecewise-smooth Van der Pol systems, where an extremum of the critical manifold arises from the nonsmoothness. In bo...

Canards, Folded Nodes, and Mixed-Mode Oscillations in Piecewise-Linear Slow-Fast Systems

Canard-induced phenomena have been extensively studied in the last three decades, both from the mathematical and from the application viewpoints. Canards in slow-fast systems with (at least) two slow variables, especially near folded-node singularities, give an essential generating mechanism for Mixed-Mode oscillations (MMOs) in the framework of smooth multiple timescale systems. There is a wea...

Mixed-Mode Oscillations in a Stochastic, Piecewise-Linear System

We analyze a piecewise-linear FitzHugh-Nagumo model. The system exhibits a canard near which both small amplitude and large amplitude periodic orbits exist. The addition of small noise induces mixed-mode oscillations (MMOs) in the vicinity of the canard point. We determine the effect of each model parameter on the stochastically driven MMOs. In particular we show that any parameter variation (s...

Canard-Like Explosion of Limit Cycles in Two-Dimensional Piecewise-Linear Models of FitzHugh-Nagumo Type

We investigate the mechanism of abrupt transition between small and large amplitude oscillationsin fast-slow piecewise-linear (PWL) models of FitzHugh-Nagumo (FHN) type. In the context ofneuroscience, these oscillatory regimes correspond to subthreshold oscillations and action potentials(spikes) respectively. The minimal model that shows such phenomenon has a cubic-like nullclin...

Piecewise Constant Controlled Linear Fuzzy Systems