Robust Sparse Identification and Saddle Mediated Transport of Nonlinear Dynamics

Abstract

The data-driven modeling approach has become increasingly popular to model the system where the first-principled approach is prohibitively challenging, such as finance, climate science, biology, epidemiology, etc. The Sparse Identification of Nonlinear Dynamics algorithm (SINDy) is a data-driven method that serves as an alternative to the first-principled modeling approach. The SINDy has the unique characteristic of identifying a sparse and interpretable model given the measurement data of the system of interest. However, the original SINDy can not identify implicit dynamics, limiting its ability to identify a dynamical system with rational functions. Moreover, common with many data-driven techniques, the SINDy is not robust to noise. This thesis proposes two important extensions that extend SINDy's ability to identify rational dynamics robustly (SINDy-PI) and improves SINDy's noise robustness by using state-of-the-art time-stepping constraints and automatic differentiation techniques (modified SINDy). We will also show SINDy's application in discrepancy modeling using various numerical examples and an experimental double pendulum. This discrepancy modeling framework of SINDy is advantageous when an imperfect model is already known. The work on the SINDy algorithm naturally brings out another important topic: how to design an experimental benchmark system for chaos, learning, and control. As a well-known classic textbook example, the multi-link pendulum on the cart system has been well studied. However, extensive documentation on such a system's design, construction, and operation is missing from the literature. Thus, this thesis also provides a detailed tutorial on building a multi-link pendulum experimental system that is highly flexible, enabling a wide range of benchmark problems in dynamical systems modeling, system identification and learning, and control. The building of the pendulum hardware system is also for the future experimental validation of the saddle-mediated transport of the double pendulum. The transport of material in chaotic dynamical systems has been shown to be mediated by the presence of saddle points in the phase space. This perspective has been used to explain efficient bio-locomotion, develop energy-efficient transport routes in the solar system for space mission design, and understand chemical reaction kinetics. In particular, the mechanical double pendulum has three fixed saddle points and exhibits rich, chaotic motion. This thesis then provides a detailed analysis to establish a correspondence between the double pendulum and the planar circular restricted three-body problem. Next, we demonstrate numerically the existence of a family of unstable periodic orbits (UPOs) that organize the phase space trajectories. This involves a thorough characterization of the homoclinic and heteroclinic connections. These UPOs and connections establish a skeleton that enables high-level control of the itinerary or topology of a future trajectory in symbolic dynamics. Those numerical analyses on the double pendulum then serve as foundational work for the future experimental verification of the saddle-mediated transport of the double pendulum using the experimental system we built.