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 (such as a modification of the piecewise-linear function in the model) that leaves the ratio of noise amplitude to time-scale separation unchanged typically has little effect on the width of the interval of the primary bifurcation parameter over which MMOs occur. In that sense, the MMOs are robust. Furthermore we show that the piecewise-linear model exhibits MMOs more readily than the classical FitzHugh-Nagumo model for which a cubic polynomial is the only nonlinearity. By studying a piecewise-linear model we are able to explain results using analytical expressions and compare these with numerical investigations.Comment: 25 pages, 10 figure

Analytical study of mixed-mode oscillations in human β-cells

β-cells are specific cells located in pancreatic islets that produce and secrete insulin. The secretion of insulin happens as a consequence of a electrical activity: in the present thesis we analyse a mathematical model regulating that phenomenon. This model depends on eight variables, which are recognized having different time scales and hence classified as slow, medium or fast. By varying some parameters of the model we remark the generation of mixedmode oscillations: small amplitude oscillations with a global return mechanism. In order to explain it, a reduction of the model is performed and a 3d-model is obtained. The study of the singularities of this system let us notice the appearance of mathematical objects called canards in their neighborhood. The strong canard determines the funnel where simulations have to enter in order to begin oscillating around the weak canard. Finally secondary canards induce a discretization of the space and determine the number of small amplitude oscillations of the solution of the system. We find good agreement between our analytical studies and numerical simulations

Tourbillion in the phase space of the Bray-Liebhafsky nonlinear oscillatory reaction and related multiple-time-scale model

The mixed-mode dynamical states found experimentally in the concentration phase space of the iodate catalyzed hydrogen peroxide decomposition (The Bray-Liebhafsky oscillatory reaction) are discussed theoretically in a related multiple-time-scale model, from the viewpoint of tourbillion. With aim to explain the mixed-mode oscillations obtained by numerical simulations of the various dynamical states of a model for the Bray-Liebhafsky reaction under CSTR conditions, the folded singularity points on the critical manifold of the full system and Andronov-Hopf bifurcation of the fast subsystem are calculated. The interaction between those singularities causes occurrence of tourbillion structure