Choreographies in a high-dimensional self-consistent transport map model

In this work we study the self-consistent chaotic transport of the vorticity in a two dimensional inviscid shear flow. In an active transport problem the advected field determines the velocity field through a dynamical constrain. The advection-diffusion equation can be reduced to a finite dimensional dynamical system given by an areapreserving self-consistent map obtained from a space-time discretization of the SingleWave model [1]. In our case, the map is defined by a large set of area-preserving twist maps which are coupled by a mean field given by the phase average of these maps. In this work, we study the impact of periodic orbits in chaotic transport and coherent structures of the self-consistent map. To justify the importance of the periodic orbits, a non autonomous map were introduced to mimic the asymptotic behavior of the selfconsistent map and its simulations suggested that transport could be associated to periodic oscillations in the mean field. Our goal is to study a special family of periodic orbits. With these orbits, that we call choreographic orbits, we can simplify a lot the problem of finding periodic orbits when the dimension N of the self-consistent map is large, N ∼ 104. The idea is to use symmetry properties of these periodic orbits, such that we can start with a low-dimension periodic orbit which we can replicate in order to obtain a full periodic orbit. These choreographic orbits can be continued from the trivial case (where the twist maps are uncoupled) using numerical and asymptotic methods. In this way, we use normal forms to describe these orbits in order to find the dependence of the parameters of the advected field with the chaotic transport of the map. Numerical simulations are used to verify the prediction obtained from the asymptotic methods.