AM108 S18 Dynamical Systems

This course provides an introduction to nonlinear dynamical phenomena, with a focus on the behavior of systems described by ordinary differential equations.

Dynamical systems theory provides a framework for thinking about the time evolution of models of real-world systems. Our focus in this course is on building intuition for this geometric way of thinking. To that end, we will study stability and bifurcations in depth. In addition, the course will briefly touch on the following topics: chaos; routes to chaos and universality; approximations by maps; strange attractors; fractals. Techniques for analyzing nonlinear systems are introduced with applications to physical, chemical, and biological systems such as forced oscillators, chaotic reactions, and population dynamics.

Assignments:

Problem Set 01 (5.3 +/- 1.4 hours)
Problem Set 02 (9.6 +/- 2.9 hours)
Problem Set 03 (5.3 +/- 1.3 hours)
Problem Set 04 (8.2 +/- 2.8 hours)
Problem Set 05 (3.2 +/- 0.8 hours)
Problem Set 06 (5.9 +/- 1.7 hours)
Problem Set 07 (6.7 +/- 1.9 hours)
Problem Set 08 (4.2 +/- 2.4 hours)
Problem Set 09 (5.6 +/- 2.2 hours)
Problem Set 10 (no data)
Problem Set 11 (4.2 +/- 1.4 hours)

Video links:
Before Class 02:
Geometric methods (sine example) - 12 min
Logistic example (Links to an external site.) - 4 min
(optional) Brief summary - 3 min
Linearizing - 10 min
No linear term - 6 min
Linearization in logistic (Links to an external site.) - 3 min
Existence, uniqueness, oscillation (Links to an external site.) - 9 min

Before Class 03:
Bifurcations (Links to an external site.) - 6 min
Saddle node bifurcation (Links to an external site.) - 5 min
Bifurcation diagram (Links to an external site.) - 5 min
Example (Links to an external site.) - 13 min
Transcritical bifurcation - 8 min
Pitchfork bifurcation (Links to an external site.) - 5 min
Subcritical pitchfork (Links to an external site.) - 7 min

Before Class 04:
Dimension (Links to an external site.) - 15 min
Nondimensionalizing (Links to an external site.) - 12 min
Bistability - 11 min
Bistability and cusp catastrophe (Links to an external site.) - 12 min
Setup of an insect outbreak example - section 3.7 (Links to an external site.) - 11 min

Before Class 05:
Encoding oscillation in a 1d system - 6 min
Oscillator example (Links to an external site.) - 12 min
Bottleneck near a bifurcation - 10 min
These fireflies each have a phase (Links to an external site.) - 2 min
Phase locking of an oscillator to a reference (Links to an external site.) - 12 min

Before Class 06:
Why work in 2d? - 3 min
Vector fields with two variables (Links to an external site.) - 8 min
Trajectories cannot cross - 13 min
Phase portraits for linear systems - 13 min
Saddle points - 7 min
optional: Linear algebra review (Links to an external site.) - 7 min
optional: Eigenvectors (Links to an external site.) - 4 min

Before Class 07:
Attracting or repelling nodes (Links to an external site.) - 13 min
Attracting or repelling spirals (Links to an external site.) - 8 min
Linear center (Links to an external site.) - 5 min
Line of fixed points + Summary (Links to an external site.) - 7 min
Nonlinear systems: linearizing (Links to an external site.) - 9 min
Does linearizing work for classifying fixed points? (Links to an external site.) - 6 min
Linear centers might not actually be centers - 19 min

Before Class 08:
Example: what happens when two species compete? (Links to an external site.) - 17 min
Trying to find the global picture from local linear portraits - 14 min
A special type of systems: energy conserving systems from physics - 6 min
What are conservative systems? - 12 min
Example: analysis of a conservative system (Links to an external site.) - 16 min

Before Class 09:
Starting and ending at a saddle: homoclinic orbits - 6 min
Using an energy surface to think about conservative systems - 8 min
Conservative systems can't have attracting or repelling fixed points - 14 min
Frictionless pendulum example - 13 min
Pendulum state space is cylindrical - 6 min
Rotation of the vector field along a closed curve - 9 min
Calculating the index: examples - 8 min
Properties of the index - 20 min

Before Class 10:
Prep for in-class quiz

Before Class 11:
Index of fixed points with Det < 0 vs with Det > 0 - 6 min
Theorem: Any closed trajectory must enclose a fixed point - 8 min
Optional: Index theory on salamander arms + some topological ideas - 9 min
Isolated closed trajectories (limit cycles) - 8 min
Rule out a closed orbit with Dulac's criterion - 17 min
Note about the Dulac's argument: you likely know of the divergence theorem in 3d, which relates the integral of the divergence of a vector field over a solid surface to the flux of the vector field out of the surface. It's 2d analog is a flux form of Green's theorem.

This version of Green's theorem relates the integral of the divergence of a vector field over a 2d region to the flux of the vector field out of the curve at the boundary of the region. This divergence theorem analog is the version of Green's theorem that is used in the following video.
Why Does Dulac's criterion work? - 7 min

Before Class 12:
Showing there must be a limit cycle in some region - 15 min
Finding such a region in a polar example - 7 min
Glycolysis: a non-polar example - 15 min
optional: Additional details of the example - 10 min
What is the van der Pol oscillator? - 7 min
Changing coordinates in the system - 13 min
Drawing the van der Pol phase portrait - 8 min

Before Class 13:
For the van der Pol in the large mu limit, how does x(t) behave? - 5 min
Estimate the period of the oscillation (as a function of mu) - 8 min
Energy argument about the weakly nonlinear (small mu) case - 16 min
Types of bifurcations in 2d systems - 12 min
Saddle-node bifurcations in 2d - 13 min
Pitchfork bifurcations in 2d - 7 min

Before Class 14:
Birth of a limit cycle via a supercritical Hopf bifurcation - 13 min
Subcritical Hopf bifurcation - 19 min
Squealing brake example - 1 min
Oscillating chemical reactions - 3 min
Briggs-Rauscher oscillating reaction - 4 min
Unmixed, so that it occurs in space as well as time: Belousov-Zhabotinsky reaction - 1 min
A bifurcation of stable and unstable limit cycles - 10 min
Limit cycle collides with a saddle in a homoclinic bifurcation - 6 min

Before Class 15:
1: Saddle-node infinite period bifurcation - 11 min
2: Cardiac dynamics link + period calculation setup (Links to an external site.) - 7 min
We saw the same period calculation when we looked at the ghost of a bifurcation (see section 4.3). We found that the period scaled as $1/\sqrt{\mu}$ 1 / μ where $\mu$ μ was the distance from the bifurcation.
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)

Before Class 16:
Prep for in-class quiz

Before Class 17:
1: Introduction to Poincare maps | (7 min)
2: Limit cycle stability: Poincare map example | (12 min)
On to 3d systems! We skipped the waterwheel example, so overlook the references to it.
3: The Lorenz equations | (7 min)
4: Simple properties of the Lorenz system | (7 min)

Before Class 18:
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)

Before Class 19:
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 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)

Before Class 20:
Recall that we had just learned about choices of the parameter r where a period-2 cycle was stable and where a period-4 cycle was stable. Steve had mentioned that we are observed 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 diagram showing 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? (related to yesterday's activity) | (10 min)

Before Class 21:
Prep for evening exam

Before Class 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.pdf that 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)

Before Class 23:

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)

Before Class 24:
Submit slides for 6 minute presentation. There will be a peer feedback process during class.

Before Class 25:
Submit revised slides for 6 minute presentation. There will be presentations during class.

Sarah Iams