Repozytorium publikacji - Politechnika Gdańska

This paper considers a problem of testing, from a finite sample, a topological conjugacy of two trajectories coming from dynamical systems (X, f ) and (Y, g). More precisely, given x1, . . . , xn \subset X and y1, . . . , yn \subset Y such that xi+1 = f (xi) and yi+1 = g(yi) as well as h : X \rightarrow Y, we deliver a number of tests to check if f and g are topologically conjugated via h. The values of the tests are close to zero for systems conjugate by h and large for systems that are not. Convergence of the test values, in the case when the sample size goes to infinity, is established. We provide a number of numerical examples indicating scalability and robustness of the presented methods. In addition, we show howthe presented method gives rise to a test of sufficient embedding dimension, mentioned in Takens' embedding theorem. Our methods also apply to the situation when we are given two observables of deterministic processes, of a form of one or higher dimensional time series. In this case, their similarity can be assessed by comparing the dynamics of their Takens' reconstructions. Finally, we include a proof-of-concept study using the presented methods to search for an approximation of the homeomorphism conjugating given systems.

Autorzy

• dr inż. Justyna Signerska-Rynkowska link otwiera się w nowej karcie ,
• Paweł Dłotko link otwiera się w nowej karcie ,
• Michał Lipiński