Dynamical Systems: Flows (MAGIC059)
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This course is part of the MAGIC core.
Many problems in Applied Mathematics are nonlinear and described by nonlinear ordinary (or partial) differential equations. This course aims to introduce students to the tools and techniques needed to understand the dynamics that might arise in such systems. The emphasis will be on concepts and examples rather than theorems and proofs, and will include a brief survey of useful numerical methods and packages. Students will be invited to submit examples of their own for possible discussion.
Autumn 2016 (Monday, October 3 to Friday, December 9)
No explicit prerequisites beyond standard undergraduate mathematics.
* Definition of a flow (ordinary differential equation), invariant sets, limit sets
* The Poincaré Map
* Equilibria, linearisation, stability of equilibria, periodic orbits and other invariant sets
* Structural stability, Hartman-Grobman Theorem, stable and unstable manifolds
* Centre manifold theorem, local bifurcations of equilibria and periodic orbits,
* Birkhoff normal form transformations for equilibria
* Examples from chemistry, population dynamics, mechanics
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The assessment for this course will be via a single take-home paper in January with 2 weeks to complete and submit online. Satisfactory answers to all three questions will achieve a passing grade. You are welcome to use your notes, solutions to examples sheets, online resources, and symbolic algebra to complete the questions.
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