Symmetric Kolmogorov phase space

Kolmogorov flow was simulated extensively in the last few years [9]. Its attractors and the structure of its phase space with a forcing wavevector k = 2 were studied in detail by [1]. We focus our example on a regime in which 4 periodic orbits merge through a gluing type of bifurcation into two different periodic orbits. This happens in the interval of 16.4 < R_e < 20.

Figure a)

  

Figure: Projection of the simulation for Kolmogorov flow into a two-dimensional slice of phase space: a) at R_e = 16.6, b) at R_e = 17.5; the full and the dashed lines correspond to clockwise and counterclockwise rotating trajectories, respectively.

64#64

shows the projection of four simulations for Kolmogorov flow at R_e = 16.6. The coordinate axes are spanned by the two Fourier modes 65#65 and 66#66. We find four limit cycles whose projection is symmetric with respect to a reflection on the diagonals in Figure a). A K-L analysis of the full simulation for one of these limit cycles generates eigenvectors that allow an embedding of the periodic orbit. Acting with the group D₄ on the embedding space we notice that it is invariant under the reflection r. The corresponding fixed point subspace in Figure is the diagonal x₁ = x₂. Hence not only the projection but the whole periodic orbit is invariant as a set (but not pointwise) under the reflection r.

Figure b) shows a projection of two simulations at R_e = 17.5, a clockwise and a counterclockwise rotating trajectory. Now the attractor is invariant as a set under the four-fold rotation C₄.

None of the solutions at R_e = 16.6 and R_e = 17.7 travel in the x-direction hence we can safely ignore the continuous part of the symmetry group, SO(2), and focus our considerations on the finite part D₄. Since D₄/Z₂ = C₄ we expect a total of four limit cycles at R_e = 16.6 and since D₄/C₄ = Z₂ we expect a total of two limit cycles for R_e = 17.5. This is in fact true which can be checked by new simulations with symmetrized initial conditions (see Figure ). A K-L analysis on a single simulation at R_e = 16.6 reveals that we need 3 eigenvectors to capture 99.97 % of the total variance of the data. As expected, a Galerkin projection onto those three K-L eigenmodes, augmented by a fourth mode representing the basic Kolmogorov flow 67#67, generates an ODE which has, for 68#68 a periodic orbit with reflection symmetry. However, since the data do not contain any information about the rest of phase space, the Galerkin projection has only a single periodic orbit.

Figure a)

  

Figure: Projection of the Galerkin simulation for Kolmogorov flow into a two-dimensional slice of phase space: a) Galerkin system based on a single trajectory at R_e = 16.6 b) Galerkin system based on the symmetrized K-L eigenvectors.

69#69

shows the projection of the limit cycle generated from a single trajectory. Instead of generating a D₄-equivariant Galerkin system by increasing the dataset four-fold (|C₄| = 4) we can use Lemma 2 and simply use the 3 K-L eigenmodes, act with the symmetry group on it to get the group orbit of the eigenfunctions and perform another K-L analysis. The resulting system is 12-dimensional. Acting with any group element 70#70 onto the 12 POD modes and generating a new 12d-Galerkin system, we find it unchanged from the old one. Hence the Galerkin system is equivariant under the D₄ symmetry group as required. Since we average over the group we automatically get a solution that has the full D₄ symmetry. This solution is equivalent to the basic Kolmogorov flow 71#71. Hence we do not need to add this mode to the Galerkin projection in this case. Note that in this case the K-L analysis on the symmetrized eigenvectors is nothing more than an orthogonalization of the coordinate system.

Upon increasing the Reynolds number the equivariant 12-d Galerkin system shows a bifurcation from the four Z₂-symmetric standing waves to the two C₄-symmetric periodic orbits similar to Figure . However while the original simulation shows a homoclinic gluing bifurcation [1] the Galerkin system does not show signs of a homoclinic behavior but exhibits hysteresis. This is due to the fact that at the homoclinic bifurcation the attractor is homogeneous in the x- direction and should be represented, in addition to the POD modes in y by Fourier modes in x. However, in order to resolve the homoclinic behavior properly in a Galerkin projection we found we would need many more modes making it a system that is almost as complicated to analyze as the original PDE simulations.