Course: B5.6 Nonlinear Dynamics, Bifurcations and Chaos (2024-25)

Lecturer: Profile: Radek Erban

Course information

General Prerequisites:

There is no optional Part B course as a formal prerequisite. This course builds on the material, which appears in several mandatory Prelims and Part A courses, including courses on differential equations, calculus, probability, linear algebra, constructive mathematics, computational mathematics, dynamics, metric spaces and analysis. Problem Sheet 0 includes questions covering some relevant background material.

Course Term: Hilary

Course Lecture Information: 16 lectures

Course Weight: 1

Course Level: H

Assessment Type: Written Examination

Course Overview:

This course aims to provide an introduction to the tools and concepts of dynamical systems theory which have become a central tool of both applied and pure mathematics with applications including chemical reaction networks, celestial mechanics, mathematical biology, fluid dynamics and social sciences.

The course will focus on both ordinary differential equations and maps. It will draw examples from appropriate model systems and various application areas. The problem sheets will require basic skills in numerical computation (numerical integration and visualisation of solutions of differential equations).

Learning Outcomes:

Students will have developed a sound knowledge and appreciation of some of the tools, concepts, and computations used in the study of dynamical systems. They will also get some exposure to some modern research topics in the field.

Course Synopsis:

The first 8 lectures of this course is part of the core syllabus for the MSc in Mathematical Modelling and Scientific Computing A2 Mathematical Methods.Lectures 1-8: Discrete-time (maps) and continuous-time (differential equations) dynamical systems. Notion of flows, stability of fixed points, Lyapunov function, invariant manifolds, stable manifold theorem, notion of hyperbolicity, center manifold. Chemical reaction networks. Stable, unstable and center subspaces. Poincaré-Bendixson theorem. Periodic solutions, stable and unstable limit cycles. Introduction to bifurcation theory, covering saddle-node, transcritical, supercritical pitchfork and subcritical pitchfork bifurcations. Extended center manifold. Logistic map. Periodic points of maps. Stability of \(N\)-cycles. Period-doubling bifurcation. Sharkovsky’s theorem. Invariant distribution.

Lectures 9-16: Bifurcations of limit cycles, covering supercritical and subcritical Hopf bifurcations, saddle-node bifurcation of cycles, infinite-period (SNIC) bifurcation and homoclinic (saddle-loop) bifurcation. Oscillations in chemical reaction networks. Weekly nonlinear oscillators. Poincaré-Lindstedt method. Conservative and non-conservative systems. Liénard systems, van der Pol oscillator. Hilbert's 16th problem. Lorenz equations. Lorenz map. Poincaré section. Poincaré map. Converse of Sharkovsky's theorem. Bernoulli shift map, symbolic dynamics. Tent map. Dynamics on metric spaces, sensitive dependence on initial conditions, transitivity, conjugate maps, chaotic dynamics.

Section outline

Select section General

General

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- Select activity Announcements
  
  Announcements Forum
- Select activity Reading List
  
  Reading List URL
- Select activity Discussion Forum
  
  Discussion Forum
Select section Course Materials

Course Materials
- Select activity Problem Sheet 0
  
  Problem Sheet 0 File
  
  This is Problem Sheet 0 of course B5.6 Nonlinear Dynamics, Bifurcations and Chaos, covering revisions of some Prelims and Part A material, we will build on in our lectures.
  
  Please solve Problem Sheet 0 before the course starts.
  
  Solutions to Problem Sheet 0 will be given in our first lecture in Week 1. The slides from the first lecture will be available by clicking [here] after the lecture.
- Select activity Lecture Notes
  
  Lecture Notes File
  
  Our lectures used a combination of slides and derivations on the whiteboard. In addition to my slides, there are also six books in the Reading List of course B5.6, which the candidates could read before or after our lectures. The Reading List is discussed on slide 4.
  
  Lecture Notes are available in the form of slides which were used in our lectures.
  
  My slides are provided as a pdf file which is intended to be displayed in the full-screen mode. All animations have been included as individual pages of the pdf file (to avoid technical problems with sharing slides with embedded video files). For example, a figure illustrating the dynamics of ODEs or maps is shown for different values of parameters on multiple pages, which will look like snapshots of a video, when the slides are viewed in the full-screen mode.
- Select activity Problem Sheet 1
  
  Problem Sheet 1 Assignment
  
  Problem Sheet 1 covers material discussed in Weeks 1 and 2. It includes seven problems in Sections A, B and C.
  
  Solutions to all problems in Sections A and C are available to students in Solutions to Problem Sheet 1 section of this course website.
  
  Solutions to problems in Section B are only available to class tutors and teaching assistants on this website.
- Select activity Solutions to Problem Sheet 1
  
  Solutions to Problem Sheet 1 File
  
  Problem Sheet 1 covers material discussed in Weeks 1 and 2. It includes seven problems in Sections A, B and C. Solutions to all problems in Sections A and C are available to students in this pdf file, while written solutions to the problems in Section B are only available to class tutors.
- Select activity Problem Sheet 2
  
  Problem Sheet 2 Assignment
  
  Problem Sheet 2 covers material discussed in Weeks 3 and 4. It includes seven problems in Sections A, B and C.
  
  Solutions to all problems in Sections A and C are available to students in Solutions to Problem Sheet 2 section of this course website.
  
  Solutions to problems in Section B are only available to class tutors and teaching assistants on this website.
- Select activity Solutions to Problem Sheet 2
  
  Solutions to Problem Sheet 2 File
  
  Problem Sheet 2 covers material discussed in Weeks 3 and 4. It includes seven problems in Sections A, B and C. Solutions to all problems in Sections A and C are available to students in this pdf file, while written solutions to the problems in Section B are only available to class tutors.
- Select activity Problem Sheet 3
  
  Problem Sheet 3 Assignment
  
  Problem Sheet 3 covers material discussed in Weeks 5 and 6. It includes seven problems in Sections A, B and C.
  
  Solutions to all problems in Sections A and C are available to students in Solutions to Problem Sheet 3 section of this course website.
  
  Solutions to problems in Section B are only available to class tutors and teaching assistants on this website.
- Select activity Solutions to Problem Sheet 3
  
  Solutions to Problem Sheet 3 File
  
  Problem Sheet 3 covers material discussed in Weeks 5 and 6. It includes seven problems in Sections A, B and C. Solutions to all problems in Sections A and C are available to students in this pdf file, while written solutions to the problems in Section B are only available to class tutors.
- Select activity Problem Sheet 4
  
  Problem Sheet 4 Assignment
  
  Problem Sheet 4 covers material discussed in Weeks 7 and 8. It includes seven problems in Sections A, B and C.
  
  Solutions to all problems in Sections A and C arbe available to students in Solutions to Problem Sheet 4 section of this course website.
  
  Solutions to problems in Section B are only available to class tutors and teaching assistants on this website.
- Select activity Solutions to Problem Sheet 4
  
  Solutions to Problem Sheet 4 File
  
  Problem Sheet 4 covers material discussed in Weeks 7 and 8. It includes seven problems in Sections A, B and C. Solutions to all problems in Sections A and C are available to students in this pdf file, while written solutions to the problems in Section B are only available to class tutors.
Select section Intercollegiate Classes

Intercollegiate Classes
- Select activity Class Signup
  
  Class Signup Class registration
  
  Registration start: Monday, 13 January 2025, 12:00 PM
  
  Registration end: Friday, 14 February 2025, 12:00 PM
- Select activity Class Tutor's Comments
  
  Class Tutor's Comments Assignment
  
  Class tutors will use this activity to provide overall feedback to students at the end of the course.
  
  Not available unless: You have signed up to a class

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