Two Dynamical System Models Based on Real-World Scenarios: A Swarming Control Model and a Surface Tension Model
CALIFORNIA UNIV LOS ANGELES
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Dynamical systems are quite often used to describe complex real-life phenomena. In this dissertation we consider two different scenarios where we propose such models. In the first part we consider the problem of collaborative searching where agents try to search for unknown targets while keeping group formation. This scenario is observed in many animal groups, and can be applied to man-made problems like searching for mines. We use a basic swarming model combined with group decision control for this scenario. We also derive some physical scaling properties of the system and compare the results to the data from the simulations. In the second part we consider a model for the droplet breakup phenomenon. The most important problem of this scenario is how to model surface tension. We explore two different models of the diffuse interface type to describe this scenario namely the Cahn-Hilliard model and the Allen-Cahn model with advection. Using asymptotic methods we correctly predict the breakup condition for the Cahn-Hilliard model. Moreover, we prove that the Allen-Cahn model will not break up under certain circumstances due to a maximum principle. Simulations in one two, and three dimensions verify the theoretical results and provide more insight into the dynamics.
- Fluid Mechanics