Extraction of the corresponding permutation matrix. The initial tour was transformed into the optimal tour by the gradient flow.[]{data-label="fig:P flow"}](P6){width="80.00000%"}

The dynamical system without constraints converges to equilibria that are given by the Procrustes solutions. To shed light on the stability and local dynamics around the optimal TSP solutions one can approximate *subsets of the stable manifold* of the Procrustes solutions such that two permutation matrices are inside these sets. This numerical study enables the analysis of the robustness of Procrustes solutions under small perturbations of the initial permutation matrix and the assessment of the ‘closeness’ the Procrustes solution is to the optimal permutation matrix. In order to compute the sets of interest, set-oriented continuation techniques developed in [@DH96] are used in [@Cit:TSP_sahai]. An example computation is depicted in Fig. \[fig:mostAttractingSets\].

![Three-dimensional projection of two subsets of the stable manifold. The omega-limit sets of a small neighborhood of the permutation matrices $P_1$ and $P_2$ form a half circle on their corresponding Procrustes set.[]{data-label="fig:mostAttractingSets"}](AttractingSet2PermutationMatrices){width=".8\textwidth"}

Moreover, one can also use set oriented methods to compute basins of attraction of optimal permutation matrices for small instances of the TSP. These basins (subsets of the stable manifold) are computed by perturbing the optimal solutions 