This hypothesis can be modeled with differential equations. By using dynamical systems methods, the catastrophe in question may be modeled by a mathematical event known as a saddle-node bifurcation. A key part of the model is the function that gives the probability of learning the northern dialect given that a fraction of the local population uses it. Other model acquisition algorithms, such as memoryless learner (Niyogi \& Berwick 1996), give the mysterious result that verb-second languages should be extremely stable, in contrast to the history of English. This new model provides an explanation for that behavior: Memoryless learners are more sensitive to noise, resulting in a differently shaped function that does not allow the northern grammar to disappear. This model demonstrates how dynamical systems theory can be used to study language change and learning models.

We begin by examining the language dynamical equation in the case where the param- eters are fully symmetric. When learning is very error prone, the population always settles at an equilibrium where all grammars are present. For more accurate learning, coherent equilibria appear, where one grammar dominates the population. We identify all bifurca- tions that take place as learning accuracy increases. This alternation between incoherence and coherence provides a mechanism for understanding how language contact can trigger change.

We then relax the symmetry assumptions, and demonstrate that the language dynami- cal equation can exhibit oscillations and chaos. Such behavior is consistent with the regular, spontaneous, and unpredictable changes observed in actual languages, and with the sensi- tivity exhibited by changes triggered by language contact. From there, we move to the extended model with multiple UGs. The first stage of analysis focuses on UGs that admit only a single grammar. These are stable, immune to invasion by other UGs with imperfect learning. They can invade a population that uses a similar grammar with a multi-grammar UG. This analysis suggests that in the distant past, human UG may have admitted more languages than it currently does, and that over time variants with more built-in information have taken over. Finally, we address a low-dimensional case of competition between two UGs, and find conditions where they are stable against one another, and where they can coexist. These results imply that evolution of UG must have been incremental, and that similar variants may coexist.

This research was conducted under the supervision of Dr. Martin A. Nowak (Program in Theoretical Biology at the Institute for Advanced Study, and Program in Applied and Computational Mathematics at Princeton University).