Grant: Discovery Grants Program - Individual

Differential equation models are at the foundation of many theoretical advances in all the sciences. Differential equations describe the rate of change of the observed quantities and solutions form a dynamical system. Symmetry is observed in many natural phenomena; it is one of the main unifying ideas in science. Symmetry can be coded in differential equation models and predictions based on these models have shown their validity and usefulness over the years. This research program focuses on the foundations and analysis of symmetric differential equations. The emphasis is on symmetry-breaking bifurcations: the study of qualitative changes in the solutions as parameters of the equations are varied with a classification of the symmetries of the solutions. Advances are sought for delay-differential equations, for which the current state of the system depends on past states; these are prevalent in models of gene networks, population dynamics, laser dynamics and many others, and on time-reversible and Hamiltonian ordinary differential equations which occur mostly in physics such as the N-body problem of celestial mechanics. The anticipated outcome of this research is a deeper understanding of the interplay of symmetry and differential equations and the discovery of new dynamical phenomena. This research is important to theoretical scientists who construct mathematical models using differential equations. It gives them tools for the analysis of their models. But more importantly, the theoretical concepts stimulate new insight and paradigms, which are important to produce a deeper level of analysis and understanding of natural phenomena. The expertise gained through this research will strengthen the Canadian school of Dynamical Systems and attract talented students to this field who will form and lead the next generation of mathematicians.