MSc Pure Mathematics and Mathematical Logic

For entry in the academic year beginning September 2027, the tuition fees are as follows:

• MSc (full-time)
  UK students (per annum): £16,500
  International, including EU, students (per annum): £39,200
• MSc (part-time)
  UK students (per annum): £8,250
  International, including EU, students (per annum): £19,600

The fees quoted above are fully inclusive of tuition, administration and computational costs.

Fees for entry are subject to yearly review. The University reserves the right to increase your tuition fee by up to 7% each year for courses lasting more than one year, including to reflect rising costs associated with delivering our educational and wider student experience. Postgraduate fees information .

Always contact the admissions team if you are unsure which fee applies to your qualification award and method of attendance.

International student CAS deposit

Self-funded international applicants are required to pay a deposit of £2500 towards their tuition fees before a confirmation of acceptance for studies (CAS) is issued. Some applicants will be required to pay a higher deposit. More information on tuition fee deposits .

All students should normally be able to complete their programme of study without incurring additional study costs over and above the tuition fee for that programme. Any unavoidable additional compulsory costs totalling more than 1% of the annual home undergraduate fee per annum, regardless of whether the programme in question is undergraduate or postgraduate taught, will be made clear to you at the point of application. Further information can be found in the University's Policy on additional costs incurred by students on undergraduate and postgraduate taught programmes (PDF document, 91KB).

Our Manchester Master's Bursaries are aimed at widening access to master's courses by removing barriers to postgraduate education for students from underrepresented groups.

We also welcome the best and brightest international students each year and reward excellence with a number of merit-based scholarships. See our range of master’s scholarships for international students .

If you have completed or are in your final year of an undergraduate degree at The University of Manchester, you may be eligible for a discount of 10% on tuition fees for postgraduate study. Find out if you're eligible and how to apply .

The Martingale Foundation helps provide access to postgraduate mathematics study for UK students facing financial barriers by offering fully-funded MSc and PhD programmes. Find more information on the Martingale Foundation website . Applications for 2027 scholarships open Wednesday 12 August 2026.

For more information on master's tuition fees and studying costs, visit the University of Manchester funding for master's courses website.

Dynamical systems theory is the mathematical theory of time-varying systems; it is used in the modelling of a wide range of physical, biological, engineering, economic and other phenomena. This module presents a broad introduction to the area, with emphasis on those aspects important in the modelling and simulation of systems. General dynamical systems are described, along with the most basic sorts of behaviour that they can show. The dynamical systems most commonly encountered in applications are formed from sets of differential equations, and these are described, including some practical aspects of their simulation. The most regular kinds of behaviour---equilibrium and periodic---are the most easy to analyze theoretically; linearization about such trajectories are discussed (for periodic behaviour this is done using the Poincaré map.)

Much more complex behaviours, including chaos, may be found; these are described by means of their attractors. The linearization approach can be extended to these, and leads to the concept of Lyapunov exponents.

In applications it is often important to know how the observed behaviour changes with changes in the system parameters; such changes can often be sudden, but frequently conform to one of a relatively small number of scenarios: the study of these forms the subject of bifurcation theory. The simplest bifurcations are discussed.

Students are not permitted to take MATH45041 and MATH65041 for credit in an undergraduate programme and then a postgraduate programme.

To develop a basic understanding dynamical systems theory, particularly those aspects important in applications. To describe and illustrate how the basic behaviours found in dynamical systems may be recognized and analyzed.

 On competion of this course unit students will be able to:

• solve simple systems of ODEs or maps, and deduce the long term behaviours of the solutions
• derive qualitative properties of solutions to systems of ODEs or maps by semi-group property, conserved quantities, invariant sets or change of variables in polar coordinates
• calculate fixed points of system of ODEs, determine their linear types and sketch the phase portrait
• construct Lyapunov function to show the stability of the solutions
• apply the Poincare-Bendixson theorem to show the existence of periodic solutions and apply Floquet theory to periodic linear system
• calculate stable and unstable manifolds;
• calculate the centre manifold and classify the bifurcation types from the reduced dynamics
• calculate fixed points or periodic orbits of maps, locate and classify bifurcation points.

1. Basics. Basic concepts of dynamical systems: states, state spaces, dynamics. Discrete and continuous time systems. [1 lecture]
   Some motivating examples: (discrete): simple population models, numerical algorithms; (continuous): chemical and population kinetics, mechanical systems, electronic and biological oscillators. [1]
2. Basic features of dynamical systems. Trajectories, fixed points, periodic orbits, attractors and basins. Autonomous and non-autonomous systems. Phase portraits in the plane and higher dimensions; examples of phase portraits of 2-d and 3-d systems. [2]
3. Ordinary differential equations. Systems of first order ordinary differential equations; initial value problems, existence and uniqueness of solutions. Flows. [2]
4. Equilibria and linearization. Fixed and equilibrium points and their linearization; classification and the Hartman-Grobman theorem; invariant manifolds; examples in 2-d and 3-d. Computing equilibrium points. Stability and Liapounov functions. [5]
5. Periodic orbits and linearization. Poincaré sections and the Poincaré map. Linearization and characteristic multipliers of periodic orbits, and stability; examples. Computing periodic orbits. [3]
6. Attractors and long-term behaviour. ω-limit sets and long term behaviour. Chaotic attractors; illustrative examples. Lyapunov exponents and their computation. One-dimensional maps and simple routes to chaos (unimodal maps and Lorenz maps). Crises, chaotic transients [8]
7. Bifurcations of flows. Local bifurcations and centre manifolds, global bifurcations; examples. Computing bifurcation diagrams by continuation.[8]

• Mid-semester coursework: 20%
• End of semester examination: weighting 80%

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
 

Essential: Meiss, James D. Differential dynamical systems. Vol. 14. Siam, 2007.
Core: Stephen H. Strogatz, Nonlinear Dynamics and Chaos, Perseus Books, Cambridge, MA, USA, 1994.
Core: Stephen Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, NY, USA, second edition, 2003.
Recommended: Lynch, Stephen. Dynamical systems with applications using MATLAB. Boston: Birkhäuser, 2004.
Further reading: Kathleen T. Alligood, Tim D. Sauer and James A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer-Verlag, New York, NY, USA, 1996.
Futher reading: Y.A. Kuznetsov Elements of Applied Bifurcation Theory, Springer Applied Math. Sci. 112, 1995.