AM108 S19 Dynamical Systems

Problem Sets

Problem Set 01 (6.2 +/- 2.2 hours)
Problem Set 02 (8.7 +/- 2.5 hours)
Problem Set 03 (7.1 +/- 2.0 hours)
Problem Set 04 (7.9 +/- 2.4 hours)
Problem Set 05 (4.3 +/- 1.7 hours)
Problem Set 06 (6.5 +/- 2.5 hours)
Problem Set 07 (7.5 +/- 1.9 hours)
Problem Set 08 (4.5 +/- 1.6 hours)
Problem Set 09 (6.4 +/- 2.0 hours)

Video assignments

Video list

Class 29: Wednesday, April 24

In class 22 we looked at the structure of fractal attractors. Now we are returning to the structure of the orbit diagram of the logistic map
1:  Brief recap + same orbit diagram of logistic map in other systems | (7 min)
2: (optional)  Recall the self similar structure of the orbit diagram of the logistic map | (3 min)
What is universal about the period doubling in the orbit diagram of the logistic map?
3:  Universal aspects of period doubling: stories about Feigenbaum | (16 min)
4:  Universal aspects of period doubling: Feigenbaum's constant delta | (8 min)
5:  More universal scaling   |   (5 min)
6:  What is a superstable fixed point? | (8 min)
7:  Superstable p-cycles | (5 min)
8:  Superstable p-cycles in the figtree diagram | (6 min)

Class 28: Monday, April 22

We are returning to the Lorenz system (and to the logistic map) to look at the geometry of the strange attractor.
1:  What is the geometry of the strange attractor? | (13 min)
Scans of the drawings of the Rössler attractor AbrahamShaw1983Dynamics.pdfthat Steve is showing on the projector
2:  Looking at the geometry of the Rössler attractor | (14 min)
3:  Is something similar happening in the logistic map? | (2 min)
4:  How does the Rössler attractor merge back into itself? | (7 min)
5:  What is the dimension of the Lorenz attractor? | (1 min)
The next videos, looking at the Cantor set and its similarity dimension, are closely related to the class activity on Wednesday.
6:  Construction of the Cantor set | (4 min)
6:  Properties of the Cantor set | (10 min)
7:  Dimension of the Cantor set | (7 min)
(optional) Belcastro2001CantorSetContainsOneFourth.pdf

Class 26: Monday April 15th

Recall that we had watched a video showing a choice of the parameter r where a period-2 cycle was stable and one where a period-4 cycle was stable. Steve had mentioned that we are observing "period doubling" as r changes.
1:  At what values of r does the period double from 1 to 2 or from 2 to 4?   | (9 min)
2: (optional)  Watch a transition to turbulence in pipe flow. Modeling this transition is what Steve was referencing when he was talking about turbulence. | (1 min)

Take a look at the orbit diagramshowing stable structures as a function of r. Video 3 is about this diagram.
3:  The orbit diagram of the logistic map.| (9 min)
4:  How do we find a bifurcation in a map? How does period doubling occur?. | (12 min)
5:  What are the points involved in the period-2 orbit? | (10 min)
40 minutes of video

Class 24: Wednesday April 10th

1:  A definition of "attractor"    |   (13 min)
2:  A possible 4th property for an attractor |   (7 min)
3:  A definition of "strange attractor" | (1 min)
4:  Use a map to see the dynamics of the Lorenz attractor | (9 min)
5: Explore the Lorenz map via a Java applet (Lorenz map applet) or watch a screen capture of the applet | (1 min)
6:  The Lorenz "map" has an (unstable) fixed point | (7 min)
7:  Stability of the fixed point of the Lorenz map | (9 min)
8:  Map example: the logistic map | (17 min)
64 minutes

Class 23: Monday April 8th

Have the Lorenz equations from last class in front of you for this first video. What is happening in the Lorenz system?
1:  Find fixed points in the Lorenz system.| (10 min)
2:  Use a Liapunov function to show (0,0,0) is globally attracting for r<1. | (18 min)
3:  C+ and C- are stable for some range of r. Then what happens?.   | (16 min)
For r=28, you can use the sensitivity applet.(if java applets work in your browser) to see the distance between two nearby trajectories evolve in time, or you can watch a narrated screen capture:
3:  What happens to nearby trajectories in the Lorenz system? (r = 28). | (5 min)
4:  The distance between nearby trajectories grows exponentially. | (9 min)
5:  A definition of "chaos".   | (4 min)
62 minutes of video.

Class 21: Wednesday April 3rd

Class 18: Wednesday March 27th

1: Saddle-node infinite period bifurcation | (11 min)
2: Cardiac dynamics link + period calculation setup | (7 min)
We saw the same period calculation when we looked at the ghost of a bifurcation (see section 4.3).
3: How do the amplitude and frequency of limit cycles behave near these bifurcations (a chart) | (5 min)
4: Coupled oscillators: a toroidal phase space | (11 min)
5: Trajectories form periodic (closed) orbits | (8 min)
6: Trajectories do not form closed orbits | (12 min)

54 minutes of videos

Class 17: Monday March 25th

Hopf bifurcations:
1: Birth of a limit cycle via a supercritical Hopf bifurcation | (13 min)
2: Subcritical Hopf bifurcation | (19 min)
The next video is of a system that displays a subcritical Hopf bifurcation:
3: Squealing brake example | (1 min)
4: Oscillating chemical reactions | (3 min)
5: Briggs-Rauscher oscillating reaction | (4 min)
6: Unmixed, so that it occurs in space as well as time: Belousov-Zhabotinsky reaction | (1 min)
7: A bifurcation of stable and unstable limit cycles | (10 min)
8:Limit cycle collides with a saddle in a homoclinic bifurcation | (6 min)
57 minutes of videos

Class 16: Wednesday March 13th

van der Pol oscillator:
1: For the van der Pol in the large mu limit, how does x(t) behave? | (5 min)
2: Estimate the period of the oscillation (as a function of mu) | (8 min)
There is an error in the next video: At 1:04:40, the last line has A^4 while the previous line has A^2.The two lines aren't equal. The first line is wrong; the second line is correct.
3: Energy argument about the weakly nonlinear (small mu) case | (16 min)
Bifurcations:
4: Types of bifurcations in 2d systems | (12 min)
5: Saddle-node bifurcations in 2d | (13 min)
6: Pitchfork bifurcations in 2d | (7 min)
61 minutes

Class 15: Monday March 11th

Showing a limit cycles exists:
1: Showing there must be a limit cycle in some region | (15 min)
2: Finding such a region in a polar example | (7 min)
3: Glycolysis: a non-polar example | (15 min)
4: optional: Additional details of the example | (10 min)
Limit cycle example: relaxation oscillations
5: What is the van der Pol oscillator? | (7 min)
6: Changing coordinates in the system | (13 min)
7: Drawing the van der Pol phase portrait | (8 min)
65 minutes (+ 10 optional minutes)

Class 13 (V): Wednesday March 6th

1: Properties of the index | (20 min)
2: Index of fixed points with Det < 0 vs with Det > 0| (6 min)
3: Theorem: Any closed trajectory must enclose a fixed point | (8 min)
4: Optional: Index theory on salamander arms + some topological ideas | (9 min)
Closed trajectories:
5: Isolated closed trajectories (limit cycles) | (8 min)
51 minutes

Class 11 (V): Wednesday February 27th (this will be on Quiz 02, but not Quiz 01).

More on conservative systems:
1: Conservative systems can't have attracting or repelling fixed points | (14 min)
2: Frictionless pendulum example | (13 min)
3: Pendulum state space is cylindrical | (6 min)
On to index theory:
4: Rotation of the vector field along a closed curve | (9 min)
5: Calculating the index: examples | (8 min)
50 minutes.

Class 10 (V): Monday February 25th

1: Example: what happens when two species compete?| (17 min)
2: Trying to find the global picture from local linear portraits | (14 min)
3: What are conservative systems? | (12 min)
4: Example: analysis of a conservative system | (16 min)
5: Starting and ending at a saddle: homoclinic orbits| (6 min)
6: Using an energy surface to think about conservative systems | (8 min)
71 minutes of video

Class 08 (V): Wednesday February 20th

1: More linear cases: Attracting or repelling nodes| (13 min)
2: More linear cases: Attracting or repelling spirals | (8 min)
3: More linear cases: Linear center | (5 min)
4: Another linear case: Line of fixed points + Summary | (7 min)
5: Nonlinear systems: linearizing | (9 min)
6: Does linearizing work for classifying fixed points? | (6 min)
7: Linear centers might not actually be centers | (19 min)
67 minutes of video

Class 07 (V): Wednesday February 13th

Class 06 (V): Monday February 11th

The videos for Monday are a relatively short set. The videos for Wednesday do not depend on the ones for Monday (it isn't a build: they are two not-so-related topics), so I am posting the Wednesday videos early in case that is helpful for you.

Class 04 (V): Wednesday February 6th

Class 03 (V): Monday February 4th

Class 02 (V): Wednesday January 30th

The videos below were recorded by Steve Strogatz at Cornell. He is also the author of our course text.

Sarah Iams