Nonlinearity and Uncertainty in Dynamical Systems: Is this a Dual Curse, or is it Job Security?
A favorite parlor game of engineers and physicists is to talk qualitatively about "how nonlinear" the dynamics are of their current problem. Another favorite conversation topic is dealing with probabilistic aspects of particular nonlinear problems. Indeed, a great deal of entropy (statistical, thermodynamic, and conversational) is generated daily trying to accommodate nonlinearity and uncertainty. Brute-force Monte Carlo simulations are the "fall back" tool, but are expensive and impossible for real-time use. On the other hand, methods for solving the high-dimensioned partial differential equation (known as the Fokker-Planck-Kolmogorov equation, or FPK) that governs evolution of the state probability density function has not matured, especially for real-time applications. Solving this class of problems has found new impetus: Being able to compute in real time the probability of collision of orbiting objects that properly reflects orbit uncertainty. Furthermore, we need to establish probabilistic methods and devise workable strategies to avoid the most likely spacecraft collisions with more than 20,000 trackable debris objects in Earth's orbit. For example, we maneuver the space station to avoid predicted collisions several times per year. Remarkably, recent progress has led to a way to effectively solve the FPK partial differential equation, and the idea may be transferrable to analogous physical systems where dominant forces are of the inverse square type. This lecture provides a "user-friendly on-ramp" into ways to determine the evolution of the highly non-Gaussian probability density functions for orbit mechanics. We also focus on how to efficiently operate on the probability density functions associated with two or more near-intersecting orbits to determine the probability of collision. This later issue is the crucial issue in space situational awareness.
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