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 $H_0 = (2\pi )^{ - 1} \sin (2\pi x_1 )\cos (2\pi x_2 )$ is studied, as well as the evolution equations for an elliptical vortex in a three-dimensional strain flow.