Brian P. Mann

Samuel A. Moore, Brian P. Mann, Boyuan Chen

Dynamical systems theory has long provided a foundation for understanding evolving phenomena across scientific domains. Yet, the application of this theory to complex real-world systems remains challenging due to issues in mathematical modeling, nonlinearity, and high dimensionality. In this work, we introduce a data-driven computational framework to derive low-dimensional linear models for nonlinear dynamical systems directly from raw experimental data. This framework enables global stability analysis through interpretable linear models that capture the underlying system structure. Our approach employs time-delay embedding, physics-informed deep autoencoders, and annealing-based regularization to identify novel low-dimensional coordinate representations, unlocking insights across a variety of simulated and previously unstudied experimental dynamical systems. These new coordinate representations enable accurate long-horizon predictions and automatic identification of intricate invariant sets while providing empirical stability guarantees. Our method offers a promising pathway to analyze complex dynamical behaviors across fields such as physics, climate science, and engineering, with broad implications for understanding nonlinear systems in the real world.

Xue-She Wang, Brian P. Mann

Recent research efforts demonstrate that the intentional use of nonlinearity enhances the capabilities of energy harvesting systems. One of the primary challenges that arise in nonlinear harvesters is that nonlinearities can often result in multiple attractors with both desirable and undesirable responses that may co-exist. This paper presents a nonlinear energy harvester which is based on translation-to-rotational magnetic transmission and exhibits coexisting attractors with different levels of electric power output. In addition, a control method using deep reinforcement learning was proposed to realize attractor switching between coexisting attractors with constrained actuation.

Xue-She Wang, Samuel A. Moore, James D. Turner et al.

Understanding the basins of attraction (BoA) is often a paramount consideration for nonlinear systems. Most existing approaches to determining a high-resolution BoA require prior knowledge of the system's dynamical model (e.g., differential equation or point mapping for continuous systems, cell mapping for discrete systems, etc.), which allows derivation of approximate analytical solutions or parallel computing on a multi-core computer to find the BoA efficiently. However, these methods are typically impractical when the BoA must be determined experimentally or when the system's model is unknown. This paper introduces a model-free sampling method for BoA. The proposed method is based upon hybrid active learning (HAL) and is designed to find and label the "informative" samples, which efficiently determine the boundary of BoA. It consists of three primary parts: 1) additional sampling on trajectories (AST) to maximize the number of samples obtained from each simulation or experiment; 2) an active learning (AL) algorithm to exploit the local boundary of BoA; and 3) a density-based sampling (DBS) method to explore the global boundary of BoA. An example of estimating the BoA for a bistable nonlinear system is presented to show the high efficiency of our HAL sampling method.

Xue-She Wang, James D. Turner, Brian P. Mann

This paper describes an approach for attractor selection (or multi-stability control) in nonlinear dynamical systems with constrained actuation. Attractor selection is obtained using two different deep reinforcement learning methods: 1) the cross-entropy method (CEM) and 2) the deep deterministic policy gradient (DDPG) method. The framework and algorithms for applying these control methods are presented. Experiments were performed on a Duffing oscillator, as it is a classic nonlinear dynamical system with multiple attractors. Both methods achieve attractor selection under various control constraints. While these methods have nearly identical success rates, the DDPG method has the advantages of a high learning rate, low performance variance, and a smooth control approach. This study demonstrates the ability of two reinforcement learning approaches to achieve constrained attractor selection.