Heteroclinic orbits and chaotic dynamics in planar fluid flows. (English) Zbl 0656.76025

Bertozzi, Andrea Louis

SIAM J. Math. Anal. 19, No. 6, 1271-1294 (1988).

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.

MSC:

76B47	Vortex flows for incompressible inviscid fluids
37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70K99	Nonlinear dynamics in mechanics
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
54H20	Topological dynamics (MSC2010)

Keywords:

elliptical vortices; Melnikov’s method. planar Smale-Birkhoff homoclinic theorem; heteroclinic saddle; chaotic dynamics; Kelvin-Stuart cat’s eye flow; two-torus; three-dimensional strain flow