Controlling Canards Using Ideas From The Theory of Mixed-Mode Oscillations by Joseph William Durham Canards are special types of periodic orbits that are associated with a dramatic change in amplitude and period due to a very small change in a parameter. Since canards typically exist only for very small regions of parameter space, they are extremely difficult to observe experimentally. In this ...

Mixed mode oscillations (MMOs) are complex oscillatory waveforms that naturally occur in physiologically relevant dynamical processes. MMOs were studied in a model of electrical bursting in a pituitary lactotroph [34] where geometric singular perturbation theory and bifurcation analysis were combined to demonstrate that the MMOs arise from canard dynamics. In this work, we extend the analysis d...

We investigate the organization of mixed-mode oscillations in the self-coupled FitzHugh-Nagumo system. These types of oscillations can be explained as a combination of relaxation oscillations and small-amplitude oscillations controlled by canard solutions that are associated with a folded singularity on a critical manifold. The self-coupled FitzHugh-Nagumo system has a cubic critical manifold f...

The Atlantis Project is a wing-based autonomous sailing vessel, based on a Prindle-19 catamaran, that has demonstrated line following under automatic control to an accuracy of better than 0.3meters. This work details both the design of the wing section and the configuration analysis for a free-rotating, aerodynamically actuated wingsail that is used as themain propulsion system. Based on numeri...

ANDREW ROBERTS: Abrupt changes and intervening oscillations in conceptual climate models (Under the direction of Christopher K.R.T. Jones) This thesis examines the role of fast/slow dynamics in understanding the mechanisms behind oscillatory patterns found in paleoclimate data. Fast/slow systems often exhibit rapid transitions between metastable states, and understanding these transitions is im...

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...

Divisible e-cash systems allow a user to withdraw a wallet containingK coins and to spend k ≤ K coins in a single operation, respectively. Independent of the new work of Canard, Pointcheval, Sanders and Traoré (Proceedings of PKC ’15) we present a practical and secure divisible e-cash system in which the bandwidth of each protocol is constant while the system fulfills the standard security requ...

We present some results on singularly perturbed piecewise linear systems, similar to those obtained by the Geometric Singular Perturbation Theory. Unlike the differentiable case, in the piecewise linear case we obtain the global expression of the slow manifold Sε. As a result, we characterize the existence of canard orbits in such systems. Finally, we apply the above theory to a specific case w...

This paper deals with the asymptotic study of the so-called canard solutions, which arise in the study of real singularly perturbed ODEs. Starting near an attracting branch of the ”slow curve”, those solutions are crossing a turning point before following for a while a repelling branch of the ”slow curve”. Assuming that the turning point is degenerate (or non-generic), we apply a correspondence...