Heteroclinic Orbits and Chaotic Dynamics in Planar Fluid Flows

An extension of the planar Smale–Birkhoff homoclinic theorem to the case of a heteroclinic saddle connection containing a finite number of fixed points is presented. This extension is used to find chaotic dynamics present in certain time-periodic perturbations of planar fluid models. Specifically, the Kelvin–Stuart cat’s eye flow is studied, a model for a vortex pattern found in shear layers. A flow on the two-torus with Hamiltonian H₀ = (2π )- 1 sin (2π x₁ )cos (2π x₂ ) is studied, as well as the evolution equations for an elliptical vortex in a three-dimensional strain flow.