Research Topic

Goals, motivation, and core questions: Dynamical systems exhibit many strange behaviors due to their complexity and the interplay of multi-physics phenomena. One such behavior that is unique to nonlinear systems is known as a bifurcation. This occurs when the system behavior qualitatively changes whether it be from fixed point stability to periodic, or periodic to chaotic, etc. Dynamical system bifurcations have been observed in many real-world systems such as the human heart causing cardiac arrythmias, aircraft upset dynamics and during epileptic seizures. These systems are all cases that are driven by many different physical laws that can introduce multiple time scales into the system and lead to bifurcations in the responses for different parameters. Typically models for complex systems of this nature contain many parameters that govern the bifurcations that can be detrimental to the operation of the system. Thus, it is important to understand where the bifurcations occur in the parameter space and know how to safely move through this high dimensional space while avoiding regions where the dynamics are unfavorable. A random walk through the parameter space could yield many bifurcations that may damage the system, but if the optimal walk is chosen, the number of bifurcations could be minimized, and the system can remain in the safe operating region. Standard bifurcation analysis only allows for a single parameter to be varied at one time which yields very little information about a system with many parameters. To solve this problem, I plan to leverage the differentiability of topological persistence maps to studying constrained parameter spaces in dynamical systems. I will study the mapping of constrained parameter space regions into the persistence diagram space and utilize the inverse maps to generate optimal paths that avoid unsafe regions (conditions that result in undesirable dynamics such as chaos) in the space. Locating such a path in the parameter space will allow for a deeper understanding of the underlying physics and ensure the safety of systems with changing parameters. 

Aim 1: Develop a Framework for Generating High Dimensional Bifurcation Spaces: I will begin by extending the results in to detect bifurcations in high dimensional parameter spaces using tools from topological data analysis. Machine learning tools such as dimensionality reduction techniques will also be applied to determine if any parameters are insignificant to the overall system behavior. 

Aim 2: Moving through the bifurcation/parameter space: In order to move through the high dimensional parameter space in an optimal fashion, I will generalize the differentiability of persistence diagrams from to allow for constraints to be implemented to ensure that the system does not enter an unsafe region of the parameter space. This will allow for an optimal path to be obtained in the parameter space for safely moving between two system configurations. Learning more about these optimal paths could eventually allow for this to become an online process during the operation of a system to avoid bifurcations in real time.

Aim 3: Experimental Analysis/Validation: Finally, I will experimentally verify the proposed framework by testing on over 60 dynamical systems in the teaspoon python library. I also plan to apply these tools to experimental data collected from hall effect thrusters from the AFRL. These systems are very difficult to model in practice and changes in parameters can yield unstable breathing mode oscillations that are undesirable for the system. The data from the AFRL provides a unique opportunity to apply the tools that will be developed to a real problem and collaborate with other researchers in the process. 

Anticipated outcomes: The proposed framework naturally splits into three phases where the first phase deals with developing high dimensional bifurcation spaces, the second phase focuses on how we can move through the space in an optimal fashion, and the final phase will be testing the methods on numerous systems. With the help of the NRT funding, I will be enabled to publish multiple works on each phase of the project. Further, software will also be written at each stage that can be made accessible for other researchers or industrial applications for navigating constrained parameter spaces. It is anticipated that this project will take two years to complete,