Applications of dynamical systems in dissipative mechanics and in topological data analysis

Abstract

•This dissertation is devoted to two distinct research endeavors on the applications of Dynamical Systems: one focuses on Dissipative Mechanics and the other on Topological Data Analysis.

•The first research investigates a simple model of a mechanical system, consisting of a rotator and a pendulum coupled via a small, time-periodic Hamiltonian perturbation and also subject to a small dissipative perturbation. The dissipative perturbation transforms the system into a non-Hamiltonian one, altering the symplectic structure to a conformally symplectic one.

•We demonstrate that this system exhibits Arnold diffusion, that is, there exist pseudo-orbits along which the energy of the rotator subsystem grows by an amount independent of the size of the coupling parameter, for all sufficiently small values of the coupling parameter. Our finding's physical significance is that, despite the dissipation effects, it is possible to overall gain a significant amount of energy over time.

•In the second research, we propose a heuristic argument for why Topological Data Analysis (TDA) is able to detect early warning signals of critical transitions in financial times series. We focus on the time period of a bubble before the tipping point. The ansatz is that the time series during such a period follows the Log-Periodic Power Law Singularity (LPPLS) model. The application of dynamical systems theory, particularly through Takens' embedding theorem and its generalization, play a role in the analysis using delay coordinate embedding.

•The outcome of our work is that whenever the LPPLS model fits the data, the TDA method produces early warning signals of critical transitions. As an illustration, we apply both the LPPLS model and TDA to Bitcoin, which has undergone numerous phases of extreme price growth and massive crashes, providing a rich dataset for exploring financial bubbles.

•Our experiments demonstrate a strong agreement between the TDA method and the LPPLS model. Moreover, even in cases where the LPPLS model fits poorly with some of the data, the TDA method still exhibit a relatively strong signal before the tipping point. This research marks the first justification of the TDA method in terms of a deterministic model for the expected dynamics of financial bubbles.

•In this dissertation, Chapters 1 through 4 are derived from the first research, while Chapters 5 through 8 are based on the second research. Some of the results presented have been published in [1] and [2].