# ACM30190 Dynamical Systems

## Academic Year 2023/2024

This module provides an introduction to the theory of dynamical systems leading up to the concept of chaos.

The course starts by considering one-dimensional flow, identifying fixed points, classifying stability, and introducing the saddle-node, transcritical and pitchfork bifurcations.

We then progress to two-dimensional flows, and discuss classification of linear systems, methods of plotting the phase plane, and limit cycles. We will consider the behaviour of conservative systems, reversible systems and Liénard systems and discuss the Poincaré-Bedixson theorem and Hopf bifurcations.

Finally, we will look at chaotic systems, and study one-dimensional maps, fractals and strange attractors.

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Curricular information is subject to change

Learning Outcomes:

On completion of this module students should be able to:
- identify fixed points of nonlinear systems.
- use linear stability analysis to classify fixed points.
- plot trajectories and phase portraits.
- discuss the various forms of stability and the relevance of conservative systems and reversible systems.
- identify bifurcation points.
- classify and describe different types of bifurcations.
- identify limit points and limit cycles and apply the Poincare Bendixson theorem, Liapunov functions and Liénard's theorem.
- discuss chaotic systems and give some examples.

Indicative Module Content:

- Flows on the line
- Fixed points and stability
- Linear stability analysis
- Bifurcations: saddle-node, pitchfork, transcritical
- Classification of linear systems
- Phase portraits
- Conservative and reversible systems
- Limit cycles
- Poincaré-Bendixson theorem
- Hopf bifurcations
- Chaos: Lorenz equations, fractals, strange attractors

Student Effort Hours:
Student Effort Type Hours
Specified Learning Activities

24

Autonomous Student Learning

40

Lectures

36

Total

100

Approaches to Teaching and Learning:
Face-to-face, on campus

Students will be assigned readings and problems in advance of lectures/tutorials. Contact hours will be reserved for interactive discussions on the assigned material and working through problems or advanced topics.

Requirements, Exclusions and Recommendations
Learning Requirements:

Dynamical systems will be studied with the aid of python code.

Students are required to be write their own python code to solve equations, calculate data and generate plots.

All students are required to have their own laptop.

Module Requisites and Incompatibles
Pre-requisite:
ACM10060 - Appl of Differential Equations, ACM10100 - Differential & Diff Equations

Additional Information:
Students must have taken ACM10100 or ACM10060

Equivalents:
Dynamical Systems (MAPH30190)

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade In Module Component Repeat Offered
Assignment: 2 problem-solving assignments Throughout the Trimester n/a Standard conversion grade scale 40% No

20

No
Class Test: Class test Week 7 n/a Standard conversion grade scale 40% No

15

No
Examination: Final exam 2 hour End of Trimester Exam Yes Standard conversion grade scale 40% No

55

No
Multiple Choice Questionnaire: Weekly quizzes Throughout the Trimester n/a Standard conversion grade scale 40% No

10

No

Carry forward of passed components
No

Resit In Terminal Exam
Spring Yes - 2 Hour
Please see Student Jargon Buster for more information about remediation types and timing.
Feedback Strategy/Strategies

• Feedback individually to students, post-assessment
• Group/class feedback, post-assessment
• Online automated feedback

How will my Feedback be Delivered?

In-class activities and weekly quizzes provide students with regular feedback. Individual feedback will be given on assignments and class tests. The assessment structure is designed such that students will receive feedback on their previous assessment prior to submitting the next.

http://www.stevenstrogatz.com/books/nonlinear-dynamics-and-chaos-with-applications-to-physics-biology-chemistry-and-engineering
Name Role
Dr Áine Byrne Lecturer / Co-Lecturer