Tools

Computer programs useful for the analysis of dynamical systems include

• Pyhton.

Lectures, Reading, Exercises and Solutions

Please find below a rough outline of the course by date. In order to be prepared for the lectures and exercise sessions, please read the chapters and try to solve the problems indicated for a given lecture.

• Monday 02.2
  Chaos and fractals; flows on the line (Chapter 1,2) (until 2.3)
  NOTE: No exercises on the 2nd.
  
  
• Wednesday 4.2
  Bifurcations in 1D (Chapters 2 and 3). As bifurcation example I go through Exam 2013, Question 1.
  Exercises: 2.1.[1-4], 2.2.[1-7;11], 2.3.2.
  Slides: Slide Hints
  Solutions: Solution


• Monday 9.2
  Bifurcations and Circle systems (Chapter 3, 4).
  Exercises: 2.4.[1;4;7], 2.5.[3-4], 2.6.2 and 2.7.6
  Solutions: Solution


• Wednesday 11.2
  Numerical Methods in Dynamical Systems, Linear systems in 2D (Chapter 5 without subchapter 5.3). Bring your laptop.
  Exercises: 3.1[1;3], 3.2.[2;4;5], 3.4.[1;3;14], 3.5.8
  Solutions: Solution


• Monday 16.2
  Linear systems in 2D (Chapter 5 without subchapter 5.3)
  Exercises: 4.1.[1-2;8], 4.3.1, 5.1.[1,9], 5.2.[2,4,11,12]
  Solutions: Hints Solution


• Wednesday 18.2
  Phase plane (Chapter 6 without subchapter 6.8) and beginning of Limit cycles (Chapter 7)
  Exercises: 6.1.[2;5], 6.3.[1;4-5;8;14]
  Solutions: Hints(Updated) Solution


• Monday 23.2
  Limit cycles (Chapter 7 without subchapter 7.4). I go through Exam 2012, question 2.
  Exercises: 6.5.[1;11;13], 6.3.10, 6.6.7
  Solutions: Solution


• Wednesday 25.2
  Weakly Nonlinear Oscillations (Chapters 7.5, 7.6).
  Exercises: 7.2.[6;10], 7.3.4, Exam 2016, Question II.
  Solutions: Solution Jupiter_example help


• Monday 02.3
  2d Bifurcations, Hopf bifurcations (Chapter 8 without subchapter 8.3, 8.4 and 8.5). I go through Exam 2019, question II
  Exercises: Look at ex. 7.3.4. again. Since there exists a closed orbit in the phase plane you should be able to find a ''trapping region'', R (The Poincaré-Bendixson Theorem). Find such a region. Also do ex: 7.6.[5;9], 8.1.6.
  Solutions: Solution


• Wednesday 4.3
  Chapter 8.6, 8.7 and Lorenz Equations (Chapter 9)
  Exercises: 8.2.1, 8.2.9, 8.3.1, 8.7.1
  Solutions: Solution Note: A small error in Jacobian, entry (1,1) should be b-1


• Monday 9.3
  Lorenz Equations (Chapter 9) and 1D maps (Chapter 10 without subchapter 10.7).
  Exercises: 9.1.4, 9.2.1, 9.2.6, 9.4.2
  Solutions: Solution


• Wednesday 11.3
  1D maps and period doubling (Chapter 10 without subchapter 10.7). I go through Exam 2020, question 3.
  Exercises: 10.1.10, 10.1.12, 10.3.2, 10.3.10
  Solutions: Solution


• Monday 16.3
  Fractals (Chapter 11)
  Exercises: 10.4.3, 10.5.1, 11.2.1, Exam 2011 ex III, 8.4.5
  Exam 2011
  Solutions: Solution
  
  
• Wednesday 18.3
  Strange attractors (Chapter 12 without subchapter 12.5)
  Exercises: 11.3.8, 11.4.2, 8.2.3 and Exam 2012
  Exam 2012, Exam 2016
  Solutions: Solution


• Monday 23.3
  Exercises:
  Time 15.15-17, Exam 2020, Exam 2021
  
  
• Wednesday 25.3
  Exercises:
  Time 13.15-15, Exam 2022, Exam 2023, Exam 2024
  
  Written exam, four hours, 8 April 2026
  
  Question hour for Exam, 31 March 2026, 13.15-14.15.