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  • Module description and Key Textbook


    • Module Description

      Brief description about this module (both MTH744P and MTH744U):

      The module develops an alternative approach to the study of dynamical systems which evolve with time, namely, it is  the qualitative approach which enables broader conclusions to be assembled to describe the behaviour of a system over long time periods.  The underlying mathematics is calculus and linear algebra.

      MODULE ASSESSMENT

      The module is assessed by a mid-term test in the form of 
      • a quiz which contributes 20% of the overall mark. and 
      • a final examination which contributes 80% of the overall mark. 
      For those who fail the module, the reassessment will be synoptic, i.e., based 100% on a resit examination in late summer at the end of the academic year.

      Lecture  and tutorial seminar  times

      The complete set of lecture notes will exhibit the full content of the module. They will be made available section by section as the module progresses.  They will be constantly referred to during the lectures and the tutorial seminar.

      • The LECTURES will be largely oriented towards  highlighting the key ideas and techniques, and giving an alternative description and understanding of the notes.  They will be given for  Weeks 1-6 and Weeks 8-12 on Tuesday 9.00 - 11.00.
      • The TUTORIAL SEMINAR will be given Thursday 11.00 - 12 noon.  I will be available immediately afterwards for further questions.


      Module Texts
      NONLINEAR DYNAMICS and CHAOS  (SECOND EDITION) - Steven H STROGATZ (WESTVIEW PRESS, 2015)
      STRONGLY RECOMMENDED

      This is an excellent supporting text with good descriptions of the various concepts in dynamical systems, and many examples, and a wide set of exercises.  The advised practice exercises in the week-by-week content are taken from the book, and I would argue that It is a NECESSARY text to access  further appropriate  exercises for better understanding of the module.

      The Strogatz text is FREELY available  on the internet. However, you should make yourself aware of any legal constraints with using free download.

      Other texts are: 

      - Differential Equations, Dynamical Systems, and an Introduction to Chaos,  Morris Hirsch, Robert L. Devaney, and Stephen Smale

      - An Introduction To Chaotic Dynamical Systems, Robert Devaney  (CRC Press)

      - Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, J Guckenheimer, P Holmes (Springer-Verlag)

      - Elements of applied bifurcation theory, YA Kuznetsov (Springer-Verlag)

       But there are loads of texts. 

       Also explore my own texts :

      Dynamical systems: differential equations, maps, and chaotic behaviour,  D.K.Arrowsmith and C.M. Place (UG/PG) (CRC Press)

      Introduction to Dynamical Systems,  D.K.Arrowsmith and C.M. Place (advanced  to research level bifurcation theory)  (Cambridge University Press)


      Learning Aims and Outcomes

      The primary learning outcome is that you will understand and be able to use some of the key ingredients of the qualitative theory of dynamical systems.  The module requires you to use a blending of linear algebra, calculus, geometry/topology and analysis  to explore and understand the qualitative approach.  

      If you immerse yourself successfully in this advanced  module, you will have:  

      • Gained an understanding of linear and non-linear dynamical systems in one and two dimensions.
      • Understood the massive change in structure that a dynamical system can undergo as parameters in the system change. 
      • Developed a skill in using geometry, analysis , and algebra as technical components in composing phase portraits. 
      • Matured as a mathematician in understanding how different approaches to the same problem can yield wildly different insights. 




  • How to get help!

    It is vital that you engage with the lecture notes, exercises,  and quizzes, and past examination papers to gain confidence in the understanding and  techniques of the qualitative theory of ordinary differential equations.  

    There may be times when you need help and you should arrange to meet up with module organiser via email to arrange an onsite meeting (Tuesday afternoon) or arrange a personal  Teams Meeting via email with your lecturer and module lead: d.k.arrowsmith@qmul.ac.uk 



  • Early feedback questionnaire

  • Week 1 - Introduction

    The first week will be a recall of ordinary differential equations (ODES), the different types, and their role in modelling.  The qualitative approach to studying ODES will be introduced and contrasted with the deficiencies in understanding an ODE by just "solving the equation and finding a solution" 

    • Week 1 - lectures 1-2: Ordinary differential equations 

      Ordinary differential equations  -properties - autonomous,  order, independent and dependent variables, qualitative and quantitative approaches - their role in modelling, dynamical systems.

      Existence and uniqueness of solutions.  Linear stability of fixed points. Phase portraits on the line. 

      Tutorial Class: Discussion of Lecture Notes  and Exercises

      Questions on the different types of ODE, the qualitative phase portrait for any system of the form \(\dot x=f(x)\), where \(x\in\mathbb R\), the real line,  and \(f:\mathbb R\rightarrow\mathbb R\). 

      Exercises 1  - STROGATZ (p37)

      2.2.1,  2.2.3,  2.2.5, 2.2.8,  2.2.9,  2.2.10

      QUIZ  0 is available 28th September, and QUIZ 1 is available 5th October. 




  • Week 2 - ODES on the real line/ Intro to Bifurcations

    Week 2 - lectures  3-4:  ODEs on the line and Bifurcations (first 3-4 pages). 

     We will consider

    -  linearisation of the system \(\dot x=f(x)\) at a fixed point \(x=x^*\), and its relevance to the local stability of the system. 

    - existence and uniqueness of solutions of ODES

    - introduction to  bifurcation theory (Section 2 of lecture notes :  Bifurcations of dynamical systems on the line (first 3-4 pages)

    Tutorial class : Discussion of exercises and lecture notes

    Exercises  1+

    Continue to attempt the following exercises 1  (Strogatz, p37) 

    2.2.1,   2.2.3,  2.2.5,  2.2.8,  2.2.9, 2.2.10  

    plus

    Exercises 2

    ' 2.4.1, 2.4.5, 2.4.7

    QUIZ  0 is available 2nd  October , and QUIZ 1 is available 5th October. 

  • Week 3 - Types of one-parameter bifurcation

    • Week 3 - lectures 5-6 : SADDLE-NODE, TRANSCRITICAL and PITCHFORK BIFURCATIONS 

      We will introduce the notion of a bifurcation diagram, and exhibit the stable one parameter families of bifurcations on the real line: the saddle-node, transcritical, and pitchfork bifurcations.  

      Tutorial class : Discussion of exercises, quizzes,  and relevant lecture notes

      Reading: Section 2.1, 2.2,2.3 of the notes

      EXAMPLES 2 on bifurcations.

      Strogatz (p80)  3.1.2, 3.1.4, 3.2.2, 3.2.4, 3.4.4, 3.4.8


  • Week 4 - Normal forms of bifurcations

    • Week 4 - Lectures 7-8 : Normal forms of 1-Parameter families of  ODES on the Line

      We will see how the normal forms for  saddle-node, transcritical, and pitchfork bifurcations arise for parametrised ODES on the real line from analysing the various cases of the Taylor expansion at a bifurcation point.  

       Tutorial class : Discussion of exercises, quizzes,  and relevant lecture notes



    • EXERCISES 2 SOLNS (3.1.2, 3.1.4, 3.2.2, 3.2.4, 3.4.4, 3.4.8) File
      1.2 MB
    • EXERCISES 2 VIDEO QMplus Media Video

  • Week 5- Dynamical systems on the circle

  • Week 6 - Linear systems on the plane - types of fixed point

    • WEEK 6 - LECTURES 11-12 : Linear systems on the plane-types of fixed point

      Linear systems will be introduced and the different types of fixed point, and their dependence on the algebraic properties of the coefficient matrix  will be investigated. 

      Tutorial class  : Will be available for discussion of previous quizzes prior the mid-term test quiz (2022) in this week 7. 



  • Week 7 - MID-TERM WEEK - PAST TESTS

  • Week 8 - Linear systems -classification

    • WEEK 8 - Lectures 13-14 : Linear systems on the plane, types of fixed point. 

      A full classification of linear systems on the plane \(\mathbb R^2\) is completed.  

      EXERCISES  4 on linear systems

      STROGATZ (p142)  5.1.2, 5.1.3, 5.1.5,  5.1.7, 5.1.9, 5.1.10 ((a),(c),(e))


    • Exercises 4 (GN) File

      STROGATZ p142

      5.1.2, 5.1.3, 5.1.5,  5.1.7, 5.1.9, 5.1.10 ((a),(c),(e))


      1.2 MB
    • QUIZ24(5) - Linear systems and phase portraits
    • QUIZ24(5) PP1-PP8 phase portraits File
      517.8 KB

  • Week 9 - Nonlinear systems I & II

    • WEEK 9 - LECTURES 15-16: PHASE PLANE AND NONLINEAR PHASE PORTRAITS, LINEARISATION  Nonlinearity,  Phase portraits, Fixed points and linearisation. 

      TUTORIAL SEMINAR  -  Linearisation examples and sketching of phase portraits. 

      QUIZ 5 IS AVAILABLE BELOW


  • Week 10 - Nonlinear systems III & IV

    • Exercise 5 Solutions File

      Strogatz p.182  (Exercises for Chapter 6 of Strogatz)

      Ex 6.3.1 

      Ex 6.3.15

      Ex 6.3.16 (a).  You can try part (b) by using StreamPlot or equivalent. 

      Ex 6.4.1

      Ex 6.5.2  Write the second order equation as a first order system in 2-variables x and y.  The "homoclinic orbit" is the trajectory that leaves a saddle point  as an unstable manifold (with increasing time) and returns to the fixed point as a stable manifold (see notes for an example). 

      Ex. Investigate the following two systems described in polar coordinates, and sketch their  phase portraits:

      (a) 

      (b) .


      1.4 MB

  • Week 11 - Nonlinear Systems V & VI

    • WEEK 11 - LECTURES 19-20:

      LIAPUNOV FUNCTIONS, LIMIT CYCLES

      Tutorial class: EXAMPLES from EXAMINATION

      EXERCISES  6   (Strogatz  p.230)  7.1.3,   7.1.4 ,  7.2.1, 7.2.2, 7.2.10, 7.2.12

    • Exercises 6 Solutions File
      362.2 KB

  • Week 12 - Nonlinear systems V & VI

    • WEEK 12 - LECTURES 21-22:  

       We will see examples of limit cycles, their importance in applications, and criteria for their existence. 

      Exercises

      7.1.3,   7.1.4 ,  7.2.1, 7.2.2, 7.2.10, 7.2.12


      Tutorial class: EXAMPLES from EXAMINATIONS


      Exercises  6         Strogatz page 230 

      7.1.3,   7.1.4       ,  7.2.1, 7.2.2, 7.2.10, 7.2.12




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  • Nonlinear(Ch5) videos

  • Assessment information

    • Assessment Pattern -  

      Format and dates for the in-term assessments - 

      Format of final assessment - 

      link to past papers - Past paper exam repository

      Description of Feedback - 

       

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