PROJECT THREE
Objective:
To look at the BEHAVIOR of a given pair of linked linear differential equations of the form …
a,b,c,d will be constants
The technique that we will be using to see the behavior of this system is a plot of the slope field also known as vector field.
This was accomplished in Matlab – if you find one vector – you can follow it’s direction so that we can see solution curves or lines.
With this graph of the field lines – it looks like the vectors are bending outward from a line through the origin. This is calledinvariant lines – it is difficult to see on this graph – but, there is another line through the origin with the same property – the (x=y) line.
This graph shows the result of not controling the length of the vectors – if we use the unit vectors – they are all the same size and will not overlap each other so that we can see the general behavior if many vectors are shown – this vector field was sparce – but is a great visual of the need for a systematic way to create these graphs.
for the linear function we create a coefficient matrix 
Different kinds of roots scenario …
Spiral sink - negative real parts and complex parts
Spiral source – positive real parts and complex parts
Saddle – one positive and one negative eigenvalue
sink – two positive eigenvalues
source – two negative eigenvalues
circular – no real parts – just complex
This was descibed as a wrinkle … it looks kind of like fabric – like a long curtain that hangs and has creases or folds in the fabric because of excess. This can also be explained as a flow on the plane.Reading the project guide – we were asked to think of a leaf following a flow in a puddle or pond. The leaf would actually flow along these vector field lines. In Linear Algebra this semester we learned about eigenvalues and eigen vectors and determinants. It was nice to see how this fit in with differential equations. Basically – by finding the roots of the polynomial – we look at a non-zero vector with neither x or y or both equal to zero.
Dr. Davis helped us create a random matrix generator in Mathematica – it saved a lot of time and energy.
Spiral source – can see the vectors coming out in a spiral from the center – positive real and complex parts…
Today – spent time understanding how to get eigenvectors and eigenvalues – looking up definitions to understand the behaviors of spiral sources and equilibrium points – found a nice picture to compliment our picture – unfortunately the electronic gremlin stole ours ….
this graph is of – one positve and one negative eigenvalue – a saddle
This is the graph of the complex solutions – no real eigenvectors …
This graph is of two negative eigenvalues… a sink.
This graph is of two positive eigen values…a source.
These are the different options that occur as far as the behaviors of the different kind of combonations of eigenvalues that you can get.
The gramlins took this graph too …
Today we were able to get graphs of the different roots – Dr. Davis showed us a technique in Mathematica that let the computer generate different matrices for us. It saved a lot of time because we had discussed just playing with matrices to get the desired results. It was much faster and more efficient to use Mathematica. We had gotten stuck trying to graph the second to last eigenvectors, Dr. Davis looked at the code and was able to see our error and explained initializing cells which was also very enlightening and useful for future endeavors.
% this is how to find the exact solutions –> dsolve(‘Dy= the equation’, ‘t’) Laptop died a little while ago – guess it was a little me and a little of the
laptop after all – I will need to input matlab solutions at a later date.
