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.

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.

Lecture 9: Example of a 2D dynamical system: the pendulum. Effects of nonlinear terms on the linear stability of fixed points.

Lecture 17: 2D maps

Lecture 18: Fundamental examples of dynamical systems: 1. the restricted 3-body problem

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)

- notes
- Stability Diagram for the imperfect bifurcation
- stabdiag_rh.m (matlab routine to plot it)
- StabilityDiagram.jpg

- Bifurcation Diagrams (h,x_star), at fixed r

- bifdiag_hx.m (matlab routine to plot it - can do any value of r)
- bifdiag1.jpg (plot for r =1)
- bifdiag2.jpg (plot for r = 0)
- bifdiag3.jpg (plot for r = -1)
- Bifurcation Diagrams (r,x_star), at fixed h
- bifdiag_rx.m (matlab routine to plot it - can do any value of h)
- bifdiag4.jpg (plot for h =0.2)
- bifdiag5.jpg (plot for h = 0.02)
- bifdiag6.jpg (plot for h = -0.2)

- notes
- Stability Diagram for insect outbreak model

- budworm_stabdiag.m
(matlab routine to create plot)

- budworm_stabdiag.jpg
- Movies for phase portraits
- budworm1.m (fixed k =
2.5, r varies)

- budworm2.m (fixed k = 8,
r varies)

- budworm3.m (fixed r = 0.4, k varies)
- budworm4.m (fixed r = 1, k varies)

- notes
- Velocity field arrow-plot for harmonic oscillator
- harmonic_flow.m (matlab routing to create arrow-plot)
- harmonic_flow.jpg (arrow plot)

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...

- 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)

- notes
- Matlab routines for the Van der Pol oscillator (vdp.m, vdp_movie.m)

- Matlab routine for the example in polar coordinates (polar.m, polar_movie.m)

- 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.

- 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)

- 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)

- 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 17: 2D maps

Lecture 18: Fundamental examples of dynamical systems: 1. the restricted 3-body problem

- notes
- Matlab routines for creating the plots
- (to come)

- 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)