Two Results on Coherent Structure Transformations in the Medium

The present paper continues the study initiated in [4] and is devoted to transformations of coherent structures in the medium of existents. We introduce the notions of a transformer , a contracting transformer with constant , the track of an existent and the track of a cohort, structural contraction, and the coherence coefficient. We also introduce the notion of a bi-Lipschitz structural contraction with constants , where and . The paper contains two results. The first result (Theorem 1): for a contracting transformer in the complete metric space of the temporal slice, we prove the existence and uniqueness of an invariant existent (a point on which the transformer does not act: ); the track of any initial existent converges to with an a priori estimate of the convergence rate; is Lyapunov stable. As corollaries, we establish the convergence of cohorts to in the Hausdorff metric and the uniqueness of the limit cohort. The second result (Theorem 2): for a bi-Lipschitz structural contraction, we prove that the image of a coherent cohort with coherence coefficient remains coherent with radii .