Controlling Canards Using Ideas from the Theory of Mixed-mode Oscillations Controlling Canards Using Ideas from the Theory of Mixed-mode Oscillations Abstract Controlling Canards Using Ideas from the Theory of Mixed-mode Oscillations

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 thesis we present a continuous feedback control mechanism which uses only the instantaneous position of the system in phase space to tune a system parameter to a value for which a canard exists. This involves controlling a slow variable to drift toward the canard parameter region, much as is the case for mixed-mode oscillations. We apply this to tune the FitzHugh-Nagumo model to produce maximal canard orbits. A system tuned to be at a parameter value where a canard exists could serve as a sensor which could detect extremely small parameter changes. When the controller is improperly configured, periodic or chaotic mixed-mode oscillations are found. We also investigate the effects of noise on this control mechanism.