Quasi-periodic oscillations, resonance, and chaos in a two-dimensional square-vortex flow

We present a numerical study of transition to chaos in a laterally bounded two-dimensional flow composed of an array of square vortices. The flow at low Reynolds numbers is invariant under two-fold reflection symmetries. We show that the flow undergoes a sequence of hopf bifurcations leading to the formation of a spatially asymmetric temporally quasi-periodic solution (a 2-torus in phase space) which remains stable inside a narrow O(1) window of Reynolds numbers. Computing the intersections of this 2-torus with a Poincare section, we identify very narrow windows O(0.1) in Reynolds number where the dynamics turn periodic due to resonance. Finally, we show that the flow transitions to chaos via the breaking-up of the 2-torus.