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.

Course meetings | 1-2:30pm Mon/Wed | Pierce Hall 301

Instructor | Sarah Iams | Pierce Hall 287 | siams@seas.harvard.edu | 617-495-5935

Teaching Staff | Alan Estrada and Eric Sun

Text and Resources | Strogatz: Nonlinear Dynamics and Chaos (1st/2nd ed) | dfield&pplane

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 / μ where 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.