Chaos in symmetric Hamiltonians applied to some exact solutions of the semi-geostrophic approximation of 2D Incompressible Euler equations

Certain symmetry properties of Hamiltonian systems possessing hyperbolic fixed points with homoclinic and heteroclinic saddle connections are exploited to conclude chaotic dynamics are present under time periodic perturbations. Specifically, the theorems are applied to a set of exact solutions to the semi-geostrophic equations in an elliptical elliptical tank. Introduction We start this paper off by giving a brief introduction to chaos so that it is clear in the subsequent section what is meant by a chaotic system. We do this by finding a reference topological space that we define to be chaotic, and defining any other system to be chaotic if it is topologically equivalent to this space. We then proceed in section 2 to conclude that Hamiltonian systems that possess certain symmetries allow us to conclude chaotic dynamics are present when there are homoclinic and heteroclinic saddle connections to hyperbolic fixed points. We then give a brief introduction to the semi-geostrophic approximation that is used in oceanography and meteorology to make weather predictions. Finally, we apply the theorems developed in section 2 to a set of exact solutions to the semi-geostrohpic approximation of the 2D incompressible euler equations to conclude chaotic dynamics are present. 1 Chaos and Nonlinear Dynamics 1.1 Topological basis for chaos Consider the following topological space. Let Σ2 denote the set of all biinfinite sequences s = (. . . , s−2, s−1, s0, s1, s2, . . .), si ∈ {1, 2}. To define a topology we need the notion of open sets. We do this by defining a metric on Σ2, so that (Σ2, d) is a metric space. We define d as,