Last modified: 22 Jul 2026 13:42
This course covers the fundamentals of dynamical systems. Often no analytical solutions exist for such systems, particularly when they are nonlinear, yet they are essential to describe many phenomena in physics, chemistry, engineering, and biology.
This course lays out the mathematical foundations required for understanding 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 |
|---|---|---|---|
| Term | Second Term | Credit Points | 15 credits (7.5 ECTS credits) |
| Campus | Aberdeen | Sustained Study | No |
| Co-ordinators |
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In this course, 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:
This course introduces topics that precedes the contents of the Level 4 Term 2 course on Nonlinear Dynamics and Chaos Theory, which will focus on bifurcations and chaos. Nonlinear Dynamics and Chaos Theory will have less emphasis on formal proof-building and more on applying concepts to build on the foundations that are established in this course.
Information on contact teaching time is available from the course guide.
| Assessment Type | Summative | Weighting | 70 | |
|---|---|---|---|---|
| Assessment Weeks | Feedback Weeks | |||
| Feedback |
Duration: 2 hours |
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| Knowledge Level | Thinking Skill | Outcome |
|---|---|---|
| Procedural | Analyse | Analyse the properties of chaotic systems and understand how they are differentiated from non-chaotic systems. |
| Procedural | Analyse | Analyse the dynamical behaviour of one-dimensional maps. |
| Procedural | Evaluate | Evaluate methods for analysing global dynamics in two-dimensional systems. |
| Procedural | Understand | Identify and classify invariant manifolds in two-dimensional systems. |
| Procedural | Understand | Understand the meaning and role of bifurcations, and classify them. |
| Assessment Type | Summative | Weighting | 15 | |
|---|---|---|---|---|
| Assessment Weeks | Feedback Weeks | |||
| Feedback |
Duration: 1 hour |
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| Knowledge Level | Thinking Skill | Outcome |
|---|---|---|
| Procedural | Understand | Understand the meaning and role of bifurcations, and classify them. |
| Procedural | Understand | Identify and classify invariant manifolds in two-dimensional systems. |
| Assessment Type | Summative | Weighting | 15 | |
|---|---|---|---|---|
| Assessment Weeks | Feedback Weeks | |||
| Feedback | ||||
| Knowledge Level | Thinking Skill | Outcome |
|---|---|---|
| Procedural | Analyse | Analyse the dynamical behaviour of one-dimensional maps. |
| Procedural | Understand | Understand the meaning and role of bifurcations, and classify them. |
There are no assessments for this course.
| Assessment Type | Summative | Weighting | ||
|---|---|---|---|---|
| Assessment Weeks | Feedback Weeks | |||
| Feedback |
Best of online exam (100%) or online exam (70%) with carried forward in-course assessment (30%) Duration: 2 hours |
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| Knowledge Level | Thinking Skill | Outcome |
|---|---|---|
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| Knowledge Level | Thinking Skill | Outcome |
|---|---|---|
| Procedural | Analyse | Analyse the dynamical behaviour of one-dimensional maps. |
| Procedural | Evaluate | Evaluate methods for analysing global dynamics in two-dimensional systems. |
| Procedural | Analyse | Analyse the properties of chaotic systems and understand how they are differentiated from non-chaotic systems. |
| Procedural | Understand | Understand the meaning and role of bifurcations, and classify them. |
| Procedural | Understand | Identify and classify invariant manifolds in two-dimensional systems. |
| Procedural | Understand | Understand the properties of fractals and how they are linked to chaotic dynamics. |
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