Georgia Institute of Technology College of Sciences

My goal is to present a computer assisted proof of a non-trivial theorem in nonlinear dynamics, in full detail.  My (quite biased) definition of non-trivial is that there should be some infinite dimensional complications.  However, since I want to go through all the details, I need these complications to be as simple as possible.  So, I'll consider the Henon map, and prove that some 1 dimensional stable and unstable manifolds attached to a hyperbolic fixed point intersect transversally.  By Smale's theorem, this implies the existence of chaotic motions.  Recall that one can prove the existence chaotic dynamics for the Henon map more or less by hand using topological methods.  Yet transverse intersection of the manifolds is a stronger statement, and moreover the method I'll discuss generalizes to much more sophisticated examples where pen-and-paper fail.

The idea of the proof is to develop a high order polynomial expansion of the stable/unstable manifolds of the fixed point, to prove an a-posteriori theorem about the convergence and truncation error bounds for this expansion, and to check the hypotheses of this theorem using the computer.  All of this relies on the parameterization method of Cabre, Fontich, and de la Llave, and on finite numerical calculations using interval arithmetic to manage the inevitable roundoff errors. Once global enough representations of the local invariant manifolds are obtained and equipped with mathematically rigorous error bounds, it is a finite dimensional problem to establish that the manifolds intersect transversally.  

Traditional spacecraft trajectory optimization approaches focus on automatizing solution generation by capturing the solution space analytically, or numerically, in a single or few instances. However, critical human-computer interactions within optimization processes are almost always disregarded, and they are not well understood. In fact, human intervention spans across the entire optimization process, from the formulation of a problem that lands on known solution schemes, to the identification of an initial guess within the algorithm basin of convergence, to tuning the algorithm hyper-parameters, investigating anomalies, and parsing large databases of optimal solutions to gain insight. Vision-based interaction with sets of multi-dimensional information mitigates the complexity of several applications in astrodynamics. For example, visual-based processes are key to understanding solution space topology for orbit mechanics (e.g., Poincare’ maps), formulating higher quality initial trajectory guesses for early mission design studies, and investigating six-degree-of-freedom (6DOF) dynamics for proximity operations. The capillary diffusion of visual-based data interaction processes throughout astrodynamics has motivated the creation of virtual reality (VR) technology to facilitate scientific discovery since the advent of modern computers. The recent appearance of small, portable, and affordable devices may be a tipping point to advance astrodynamics applications via VR technology. Nonetheless, the tangible benefits for adoption of virtual reality frameworks are not yet fully characterized in the context of astrodynamics applications. What new opportunities virtual reality opens for astrodynamics? What applications benefits from virtual reality frameworks? To answer these and similar questions, our work focuses on a programmatic early assessment and exploration of VR technology for astrodynamics applications. The assessment is constructed by a review of VR literature with elements that are external to the astrodynamics community to facilitate cross-pollination of ideas. Next, the Johnson-Lindenstrauss lemma, together with a set of simplifying assumptions, is employed to analytically capture the value of projecting higher-dimensional information to a given lower dimensional space. Finally, two astrodynamics applications are presented to display solutions that are primarily enabled by virtual reality technology.

Living and active systems exhibit various emergent dynamics necessary for system regulation, growth, and motility. However, how robust dynamics arises from stochastic components remains unclear. Towards understanding this, I develop topological theories that support robust edge states, effectively reducing the system dynamics to a lower-dimensional subspace. In particular, I will introduce stochastic networks in molecular configuration space that enable different phenomena from a global clock, stochastic growth and shrinkage, to synchronization. These out-of-equilibrium systems further possess uniquely non-Hermitian features such as exceptional points and vorticity. More broadly, my work  provides a blueprint for the design and control of novel and robust function in correlated and active systems.