LECTURE
NOTES
Scanned and powerpoint lecture notes.
WARNING: downloading the lecture notes and putting them under your
pillow at night will not help you learn the material.
Same for putting the textbook under your
pillow. I tried it for many years without success.
If
you would like to have the movies that I showed, please come to see me
to copy them directly from my computer - most of them are very large.
Lecture
1: A global overview of nonlinear dynamics and chaos
Lecture 2: Flows on the line
Lecture 3: Bifurcations (part 1).
Saddle-nodes and transcritical bifurcations. Normal forms.
- notes
- matlab files to make your own images and animations of the
bifurcations studied
- bif1.m (normal form
saddle-node)
- bif2.m (normal form
transcritical)
- bif3.m (normal form
supercritical pitchfork)
- bif4.m (normal form
subcritical pitchfork)
- bif5.m (movie for r-x and
e^(-x))
- bif6.m (movie for r-x-e^(-x))
- bif7.m (zoomed in movie for
r-x-e^(-x))
- bifdiag1.m (plot for
rotated bifurcation diagram - r as a function of x_star)
- bifdiag2.m (plot for
normal bifurcation diagram - x_star as a function of r)
Lecture 4: Bifurcations (part 2).
Pitchforks. Example of the bead on a hoop. Imperfect bifurcations.
Notion of co-dimension.
- notes
- Stability Diagram for the imperfect bifurcation
- Bifurcation Diagrams (h,x_star), at fixed r
- Bifurcation Diagrams (r,x_star), at fixed h
- bifdiag_rx.m (matlab
routine to plot it - can do any value of h)
Lecture 5 and 6: Applications
of 1D
continuous systems: Population dynamics
- notes
- Stability Diagram for insect outbreak model
- Movies for phase portraits
Lecture 6 and 7: Introduction to 2D
systems.
Linearization around fixed points.
- notes
- Velocity field arrow-plot for harmonic oscillator
Lecture 8: Solutions of linear systems
and classification of fixed points.
Lecture 9: Example of a 2D dynamical
system: the pendulum. Effects of nonlinear terms on the linear
stability of fixed points.
- notes
- Phase portrait for un-damped pendulum (pendulum_portrait.jpg)
- Matlab routines for creating undamped pendulum movies (pendulum_undamped_movie.m
and pendulum_undamped.m)
- for the first movie shown in class (oscillating pendulum), use
imax = 40, strl = 1, tspan = 0:0.005*i:i/2, axis([-1 1 -1.5 0.5]);
- for the second movie shown in class (pendulum going round and
round) use imax = 80, strl = 4, tspan = 0:0.00125*i:i/8, axis([-10 10
-10 10]);
- Matlab routines for creating the damped pendulum movies (pendulum_damped_movie.m
and pendulum_damped.m)
- for the movie shown in class, use b = 0.1, imax = 160, tspan =
0:0.0025*i:i/4, axis([-10 10 -10 10]);
- for a movie showing what happens when b=3, change the value of
b in pendulum_damped.m and explore parameters...
Lecture 10: General discussion of
structural stability. Example of the predator-prey model. Conservative
and reversible systems.
- notes
- Matlab routine for creating phase portrait of pendulum with
trajectory superimposed on lines of constant E (pendulum_undamped_portrait_movie.m)
- Matlab routine for creating movie of predator prey phase portrait
trajectories alone (predatorprey.m
and predprey.m)
- to change parameters for the model, vary the parameters a, b, c
and d in predprey.m
- Matlab routine for creating movie of solution (predatorprey2.m)
- Matlab routine for creating phase portrait of pendulum with
trajectory superimposed on lines of constant E (predatorprey_portrait.m)
Lecture 11: Limit cycles.
Lecture 12: The solar dynamo. An
example of 3D continuous systems, and an introduction to bifurcations.
- PPT presentation (PDF file).
Movies turned out to be rather large. If you would like to have them,
please bring a memory stick and download them directly from my laptop.
Lecture 13: Introduction to discrete
systems. 1D Maps.
- notes
- Matlab routines (shamelessly stolen and adapted from this website.
A million thanks to whomever wrote them)
- logistic.m (the logistic
function, to practise using the following routines)
- iterates.m (computes N
iterates of the orbit of a
discrete system. Required for other routines)
- cobweb.m (draws a cobweb
diagram)
- cobwebmovie.m
(animates the drawing of a cobweb diagram)
- bifur.m (computes a
bifurcation diagram)
Lecture 14: The Logistic Map in
detail. Universality and Period Doubling route to Chaos.
- notes
- Matlab routines (most of them not annotated, sorry, running out
of time)
- pitchfork.m (to make the
simultaneous movie of F(x) and F^2(x) for the logistic equation)
- pitchfork2.m (to make
the simultaneous movie of F(x), F^2(x) and F^4(x) illustrating
universality)
Lecture 15: Definition of Chaos.
Lyapunov exponents; examples. The Tent Map
- notes
- Matlab routines
- cobwebdifferencemovie.m
(to make movies of 2 simultaneous cobwebs - requires iterates.m and
logistic.m for instance)
- OrbitDifference.m
(adapted from same website as above) (to calculate the distance between
two orbits)
Lecture 16: From dynamical systems to
Fractals. The Tent Map in detail. Definition of Fractals.
Lecture 17: 2D maps
Lecture 18: Fundamental examples of
dynamical systems: 1. the restricted 3-body problem
- notes
- Matlab routines for creating the plots
Lecture 19: 2. The Lorenz
equations
- notes
- Matlab routines for creating the plots
- lorenz.m (right hand side
of the lorenz system, required for the other routines. Change
parameters at will)
- lorenz_movie.m
(animates the solution in the butterfly diagram)
- lyapunov_lorenz.m
(calculates the divergence of two trajectories)
- lorenzmap.m (plots the
lorenz map out of a real trajectory)
