Bifurcation theory for systems with multiple timescales
The field of singular perturbation theory has a long and venerable history in applied mathematics with its roots in the classical theory of asymptotic expansions and averaging. Recently, there has been an explosion of interest in geometric singular perturbation theory as a means of combining geometric analysis and dynamical systems theory to provide fresh insight into problems with multiple timescales (e.g. timescales differing by orders of magnitude). For this we developed a global bifurcation theory that describes the fundamental structure of the solution space. My work in singular perturbation theory was motivated by models arising in neurobiology. Models such as the well-known Hodgkin-Huxley model predict how chemical processes occurring on different timescales affect the voltage difference across the membrane of an axon of a neuron. In particular, simple models of the action potential of neurons connected by reciprocally inhibited synapses have been studied to further understand such biological phenomena as heartbeat, swimming, and feeding. Two identical oscillatory neurons connected by reciprocally inhibitory synapses will oscillate exactly out of phase of each other, that is, while one neuron is active the other is quiescent. Our goal is to use singular perturbation theory to understand solutions of a set of four singularly perturbed differential equations which model two asymmetric oscillators. As we investigated the solution space of the asymmetric problem, we found many solutions that were qualitatively similar to the solutions of the symmetric system. We also found other very different types of behavior. For instance, in a small parameter range, there exists (at least) two stable periodic orbits of the full system. Both of these periodic solutions correspond to more complicated behavior than the typical reciprocally inhibitory behavior described above. Instead of the orbit consisting of two fast transitions, the periodic orbits consist of nine and eleven fast transitions, respectively, and the behavior of the two neurons can no longer be classified simply as active or quiescent. The possible implications of the existence of two stable periodic solutions as well as the structure of these solutions are a source of continued research.In addition to bistability in the system, we also found canard solutions, that is, solutions in which part of the orbit occurs on an unstable portion of the slow manifold. We identified two different types of canard solutions. One type of canard solution consists of a fast transition to an unstable part of the slow manifold. In the other type of canard solution, the orbit continues past a fold in the slow manifold onto the unstable part of the manifold. For a small parameter range, the family of canard solutions is stable. Continuation of the two stable periodic solutions reveals that the canard solutions persist for a much larger parameter regime but are unstable. In order to further understand the solution space of this model and other like it, we have developed a mathematical theory that clarifies the structure of the solution space of these models by considering the prototypical example of the forced Van der Pol oscillator, a model for an electrical circuit with large damping coefficient. Recently, we produced a global bifurcation analysis of a related hybrid system and, in a proceeding publication, extended the results to include canard solutions, which are commonly found in models of this type but remain computationally challenging and analytically elusive.
Maintained by: Kathleen A. Hoffman
(khoffman@math.umbc.edu).