Lab #4

This worksheet does not need to be handed in, but you are responsible for the material. The ideas laid out here will be used in next week's lab (which will be graded). Some of the instructions for this lab are taken from J. C. Polking, MATLAB Manual: Ordinary Differential Equations, Prentice-Hall, 1994.

Gradients appear everywhere in nature. Newton's law of graviation involve the gradient of a ``gravitational potential function.'' The equations of electromagnetism (Maxwell's equations) and fluid dynamics (Navier-Stokes equations) also involve gradient terms. Furthermore, gradient differential equations are used in science and engineering to optimize functions of several variables. Understanding the geometry of gradients is one of the most important goals of this course.

The purpose of the lab is

• to learn how to numerically produce solutions to a planar differential equation using Matlab and PPLANE.
• to explore the geometry of gradient differential equations.
• to better understand the distinction between graphs and images of functions.

In this lab we will study gradient flows while simultaneously learning to use PPLANE. To get started, invoke matlab by typing matlab in a Unix shell window. When the matlab prompt appears, type pplane to launch the ``phase plane'' module. The PPLANE Setup window will appear.

In this lab we will examine the gradient flow for the following function:

This function contains several critical points of varying types.

The gradient flow of a function f is the differential equation defined by , . For functions of two variables, this can be rewritten as

Since the gradient vector always points in the direction in which f increases the fastest (or, more correctly, the direction in which the linear approximation to f increases the fastest), trajectories in a gradient differential equation ``flow uphill.'' You can think of the trajectories as the paths in (x,y)-space traced out by a robot that is programmed to always walk in the steepest direction.

For our example function, and . Therefore the gradient differential equation determined by the function f is

Note that for each point (x,y), the right-hand side of the differential equation, , defines a vector. We therefore say that the right-hand side of the differential equation defines a vector field.

Activity: Sketch the graph of the current function over the domain and (you may use Maple if you wish). If possible, locate critical points of the function. Identify where the function is high and where it is low.
PPLANE has a default differential equation. Delete it, and type the right-hand side of the differential equation above into the first two fields of the PPLANE Setup window. In the lower-left corner of the setup window, change the minimum and maximum values of the display window to and . Now hit the Proceed button and the PPLANE Display window will appear.

Inside this window is a representation of the vector field for this differential equation. The vector field at (x,y) must be tangent to the solution curve passing through (x,y). To compute and plot a solution curve from an initial point, move the mouse to that point, and click the left mouse button. The solution will be computed and plotted: first in the direction of increasing time, then in the direction of decreasing time.

Activity: Compute eight or ten trajectories starting at different initial conditions (This is called a ``phase portrait.'') Do the trajectories flow uphill? Can you use the trajectories to help you locate minima and maxima of the surface? Are there any critical points not ``located'' by the trajectories? If so, what do these critical points look like (max/min/saddle)?
Activity: Evaluate the gradient at each critical point that you find analytically. What is the magnitude of the gradient at a critical point?

Activity: Suppose you start a trajectory exactly at a critical point; what does the trajectory look like? Does the shape of the trajectory depend on the type of critical point? Let's test your guess. Erase all solutions by choosing the appropriate entry under the PPLANE Options menu of the PPLANE Display window. Under the same options menu, start a trajectory at the exact location of a critical point by choosing the ``Keyboard Input'' menu entry. A small window will appear; type in the location of a critical point and then press the Compute button to generate a trajectory starting from that point. (If ever a trajectory looks different than you expect, ask yourself ``who is correct'', you or the computer? When doing numerical experiments, DO NOT assume that the computer is always correct!)
Activity: For a vector field, places where the vector field is equal to the zero vector are called equilibria. Use PPLANE to help you numerically find an equilibrium point (look under the Options menu). When the vector field is the gradient field for some function, what is the relationship between ``zeros'' of the vector field and critical points of the function?
Activity: Use the ``zoom in'' feature of PPLANE to zoom in on (1) places where the vector field is not zero and (2) places where the vector field is zero. Based on your experiment, schematically represent what a gradient vector field looks like
1. away from a critical point
2. near a maxima
3. near a minima
4. near a saddle point
(To ``un-zoom,'' retype and into the lower-left corner of the setup window and hit Proceed.)
A trajectory is a parametrized curve, , such that the tangent vector to the curve at is exactly equal to the vector field evaluated at . Thus differential equations give us another opportunity to study parametrized curves.

Note that once a trajectory is numerically determined, we can view the trajectory in several ways:

1. as a parametrized curve in the (x,y)-plane. (This is what we have been doing so far today).
2. as the graph t versus x(t) and the graph t versus y(t).
3. as the graph of the function in three-dimensional space

The first point of view is the view adopted in dynamical systems, and leads to a rich geometric understanding of differential equations. This is the point of view we will typically use in this class. The second point of view is used when it is possible to explicitly solve a differential equation for the functions x(t) and y(t). We will see some of this in the next quarter of this course. The third point of view unites the previous two views, and leads to a better understanding of the difference between ``graph'' and ``image.''

Activity: Locate a trajectory that approaches a maximum. Under the Graph menu of the PPLANE Display window, select ``Both.'' PPLANE will ask you to select a trajectory; do so by clicking on the trajectory with the left mouse button. The program will display t versus x(t) and t versus y(t) on the same graph.
1. as t increases, what value does x(t) approach?
2. as t increases, what value does y(t) approach?
3. What do these two values mean geometrically?

Activity: Now change the graph to ``Composite.'' Look carefully at the axes. What is being displayed? How does this graph show the relationship between the graphs of x(t), y(t) and the image of the function that maps t to (x(t),y(t))?