B5.6 Nonlinear Dynamics, Bifurcations and Chaos (2023-24)
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- Lecturer: Profile: Radek Erban
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).
The first half of this course is part of the core syllabus for the MSc in Mathematical Modelling and Scientific Computing A2 Mathematical Methods. Synopsis items marked with * are NOT part of the MSc syllabus
- Geometry of linear systems
Basic concepts of stability and linear manifold of solutions. Orbits in phase-space, linear flows, eigenvalues of fixed points. - Nonlinear dynamics
Notion of flows, invariant sets, asymptotic sets, attractor. Conservative and Non-Conservative systems. - Local analysis
Stable manifold theorem, notion of hyperbolicity, center manifold. - Bifurcation.
Bifurcation theory: codimension one normal forms (saddle-node, pitchfork, trans-critical, *Hopf). *Poincare-Lindstedt method. - *Maps
Poincaré sections and first-return maps. Stability and periodic orbits; bifurcations of one-dimensional maps, period-doubling. - *Chaos
Maps: Logistic map, Bernoulli shift map, symbolic dynamics, Sharkovsky's theorem. Differential equations: Lorenz equations.
Section outline
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Our lectures used a combination of slides and derivations on the whiteboard. 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. In addition to my slides, there are also six books in the Reading List of course B5.6, which are discussed on pages 56, 57 and 543 of my slides.
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