Last modified: 23 Aug 2024 15:46
This two-part course covers the fundamentals of nonlinear dynamical systems. Often no analytical solutions exist for such systems, yet they are essential to describe many phenomena in physics, chemistry, engineering, and biology.
Part I lays out the mathematical foundations required for understanding nonlinear dynamical systems. The focus is on the dynamical behaviours exhibited by linear systems, how these describe nonlinear systems locally, and how these can model time varying natural systems.
| Study Type | Undergraduate | Level | 4 |
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| Term | First Term | Credit Points | 15 credits (7.5 ECTS credits) |
| Campus | Aberdeen | Sustained Study | No |
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This is the first part of a two-part course on nonlinear dynamics and chaos theory.
In part I, we describe the mathematics of dynamical systems that are differentiable, i.e. dynamical systems which can be described using systems of ordinary differential equations (ODEs).
We will first introduce the ideas of phase space and familiarise ourselves with dynamics in phase space. By visualising the dynamics in this phase space, we take a glimpse into the concepts of fixed points and trajectories, geometrically.
We then proceed to formally study these in linear systems, where normally only one fixed point is possible. We will study how the trajectory of linear systems approach (or diverge away from) this fixed point, by using techniques from linear algebra such as matrix diagonalization.
We then proceed to prove the existence and uniqueness of solutions and understand notions of stability of fixed points. 
The important link to nonlinear dynamical systems is then made by studying the idea of linearization, which suggests that the behaviour of nonlinear systems locally around fixed points is similar to linear systems. We formalise this through the Hartman-Grobman theorem.
We then introduce the ideas of limit sets, where we are introduced to other possible long-term behaviours, such as periodic orbits. This is extended further when we study attractors, and basins of attraction, which describe the regions of phase space that converge to these attractors over time. We end the course by extending ideas of stability to periodic orbits and introducing Poincaré maps.
Syllabus:
In Part II will focus on bifurcations and chaos, where there will be less emphasis on formal proof-building and more on applying the concepts to build on the foundation that Part I established.
Information on contact teaching time is available from the course guide.
| Assessment Type | Summative | Weighting | 70 | |
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| Assessment Type | Summative | Weighting | 15 | |
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| Assessment Type | Summative | Weighting | 15 | |
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There are no assessments for this course.
| Assessment Type | Summative | Weighting | ||
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Resit: Exam (2 hours). Best of (resit exam mark) or (resit exam mark with carried forward CA marks). |
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| Factual | Remember | ILOs for this course are available in the course guide |
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