CAIMS*SCMAI Doctoral Dissertation Award 2003
Winner and Abstract

Ramadan Akila, Department of Mathematics and Statistics, University of Guelph:

Hopf Bifurcation of Coupled Oscillator Systems

This study of spatio-temporal patterns in two dimensional arrays of regularly spaced oscillators with symmetric nearest neighbour coupling is motivated by the arrays of parallel circular tubes in heat exchangers subject to a uniform cross-flow. It is well known that heat exchanger arrays may undergo oscillations which lead to fatigue, wear and costly repairs. We assume that "fluidelastic instability" is the only mechanism that causes these oscillations. The analysis deals with periodic motions of the entire array rather than individual cells, and it exploits the symmetry and the geometry of the array using results from equivariant bifurcation theory. This work presents a complete list of invariants, equivariants, normal forms, isotropy subgroups and fixed-point subspaces, for the cases with spatial periodicity N = 2,3,4, both with and without a Z_2-internal symmetry, carried out for the case of a rectangular array of tubes. The analysis includes all the generic equivariant Hopf bifurcations in this setting and determines the onset of stability and the generic behavior of the patterns. We do this by examining the generic behavior using the Equivariant Hopf Bifurcation Theorem and then determining the expected solution branches in systems of two rings of coupled identical oscillators. We verify the predicted results by numerically presenting two specific examples for each case describing the equations of motions of the tubes in the array. The possible spatio-temporal patterns of motion are determined for such arrays and the mechanisms are identified for those that are most compatible with the assumed properties of a heat exchanger, and therefore of primary concern to an engineering designer.