Abstract
The guarantee that the solutions to a system of differential equations exist and are unique is one of the core principles of dynamical systems theory. Since these properties are imparted by way of the system’s smoothness, no such guarantees exist for non-smooth systems; where differentiability and smoothness of the vector field are lost at isolated singularities or across entire discontinuity surfaces. Here we study systems that exhibit well-defined solutions in smooth regions of the phase space until they encounter particular non-smooth singularities, afterwhich the future of the solution becomes non-unique, belonging to a set of different yet equally valid trajectories. The trajectories of the system are therefore deterministic everywhere except at these special points, which we call determinacy-breaking events.Three different determinacy-breaking events are studied, our ‘routes to indeterminacy’ (á la Eckmann’s ‘routes to chaos’ of the 1980s). Each one produces a set-valued solution by means of a different singularity; the analysis and characterization of which forms the foundations of this work. Furthermore we try to motivate the relevance of such events to mathematical modeling throughout; arguing that the structure of the non-unique solution set and the mechanism which created it can both be powerful tools for a modeller interested in systems whose dynamics seem ambiguous or unpredictable.
| Date of Award | 28 Nov 2019 |
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| Original language | English |
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| Supervisor | Mike R Jeffrey (Supervisor) |