Synopsis for B8b: Nonlinear Systems

Level: H-level Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code 2B47)

Recommended Prerequisites:

Part A core material (especially differential equations). The introductory Michaelmas Term course B568a is a prerequisite for both parts of the course, and the material in that course will be assumed to be known.

Overview

This course aims to provide an introduction to the tools of dynamical systems theory which are essential for the realistic modelling and study of many disciplines, including mathematical ecology and biology, fluid dynamics, granular media, mechanics, and more. The course will include the study of both nonlinear ordinary differential equations and maps. It will draw examples from appropriate model systems and various application areas. The problem sheets will require numerical computation (using programs such as Matlab).

Learning Outcomes

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

Synopsis

1. Bifurcations and Nonlinear Oscillators [8 lectures]
   1. Bifurcation theory: standard codimension one examples (saddle-node, Hopf, etc.), normal forms and codimension two examples (briefly).
   2. Non-conservative oscillators: Van der Pol's equation, limit cycles.
   3. Conservative oscillators (introduction to Hamiltonian systems): Duffing's equation, forced pendulum.
   4. Synchronization: synchronization in non-conservative oscillators, phase-only oscillators (e.g., Kuramoto model).
2. Maps [2 lectures]
   1. Stability and periodic orbits, bifurcations of one-dimensional maps.
   2. Poincaré sections and first-return maps (leads to part 3 topics)
3. Chaos in Maps and Differential Equations [4 lectures]
   1. Maps: logistic map, Bernoulli shift map, symbolic dynamics, two-dimensional maps (examples could include Henon map, Chirikov–Taylor [“standard”] map, billiard systems)
   2. Differential equations: Lyapunov exponents, chaos in conservative systems (e.g., forced pendulum, Henon–Heiles), chaos in non-conservative systems (e.g., Lorenz equations)
4. Other topics [2 lectures or as time permits] Topics will vary from year to year and could include: dynamics on networks, solitary waves, spatio-temporal chaos, quantum chaos.

Students are by no means expected to read all these sources. These are suggestions intended to be helpful.

1. S. H. Strogatz, Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry and Engineering (Westview Press, 2000).
2. E. Ott, Chaos in Dynamical Systems (Second edition, Cambridge University Press, Cambridge, 2002).
3. P. Cvitanovic, et al, Chaos: Classical and Quantum (Niels Bohr Institute, Copenhagen 2008). [Available for free online at \href{http://www.chaosbook.org/}{http://www.chaosbook.org/}]
4. R. H. Rand, Lecture Notes on Nonlinear Vibrations. [Available for free online at \href{http://audiophile.tam.cornell.edu/randdocs/nlvibe52.pdf}{http://audiophile.tam.cornell.edu/randdocs/nlvibe52.pdf}]
5. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields (Springer-Verlag, 1983).
6. G. L. Baker and J. P. Gollub, Chaotic Dynamics: An Introduction (Second edition, Cambridge University Press, Cambridge, 1996).
7. P. G. Drazin, Nonlinear Systems (Cambridge University Press, Cambridge, 1992).
8. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Second edition, Springer, 2003).
9. S. H. Strogatz, `From Kuramoto to Crawford: exploring the inset of synchronization in populations of coupled oscillators', Physica D 143 (2000) 1-20.
10. Various additional books and review articles (especially for some of the `other topics').