Differential Equation - Plotting Phase Plane

[cse63146]

Homework Statement

Plot the phase plane of: y(t) = e^-6t(2 1) + e^-3t(1 -1)

Homework Equations

The Attempt at a Solution

I know it's a sink because $$\lambda_2 < \lambda_1 < 0$$.

http://img35.imageshack.us/img35/8108/sink.th.jpg [Broken]

but the phase plane in the answer is slanted to the right. Does anyone know why?

 

[Mark44]

I'm going by memory from a lot of years ago, so I might be wrong. I seem to recall that the axes in the phase plane were the eigenvectors, which appear to be [2 1] and [1 -1] in your problem statement.

 

[cse63146]

So here's how the answer in the textbook appears (roughly):

http://img16.imageshack.us/img16/5559/sinki.th.jpg [Broken]

so the top left corner of the axis corresponds to the (2,1) eigenvector and the bottom right corner corresponds to the (1, -1) eigenvector?

 

[Mark44]

No, the axis running from the lower left to upper right corresponds to [2 1], and the other axis corresponds to [1 -1]. If I'm anywhere close to being right on this, the axis with the negative slope has a slope of -1 (like in your drawing), but the one with positive slope has a slope of 1/2 (which is less than what you show).

 

[cse63146]

I understand now. Thanks for your help.

 

[cse63146]

Sorry, got another question. If the eigenvalues were 0 and 5, would it be a source?

 

[Mark44]

I think so, but again, I studied this a long time ago, and don't have the text I used in front of me (Dynamical Systems, Steven Smale). As I recall, besides sinks and sources, you could also have a situation where the phase diagram showed circular or elliptic orbits. I don't recall offhand what the conditions were on the eigenvalues.

 

[cse63146]

that's a center and it's when you have a root bi.

 

[cse63146]

Sorry to bother you again.

for y'' + 4y' + 13y = 0

The general solution would be y(t) = k₁e^-2tcost3t + k₂e^-2tsin3t + (27/145)cos2t + (24/145)sin2t.

It's a spiral sink, so would the phase plane (with inital conditions y(0) = 1, and v(0) = -2)look like this:

http://img13.imageshack.us/img13/8230/spiralsink.th.jpg [Broken]

not sure what to do since the eigenvectors have complex numbers.

and would the y(t) v(t) graphis look like this:

http://img13.imageshack.us/img13/4392/yvgraph.th.jpg [Broken]

 

[Mark44]

I won't be able to respond to your question about the phase plane question until I can review the text I mentioned.

Until then, though, I have a question about this DE.

for y'' + 4y' + 13y = 0
The general solution would be y(t) = k1e-2tcost3t + k2e-2tsin3t + (27/145)cos2t + (24/145)sin2t.

For the DE above, the solution would be just y(t) = k₁e^-2tcos3t + k₂e^-2tsin3t. Are you showing me just the homogeneous equation? Is the nonhomogeneous equation something like this -- y'' + 4y' + 13y = cos2t ?

 

[cse63146]

Oops. You're right. it was just the homogenous part. I was looking at a different question (which the nonhomogenous part).

 

[Mark44]

Since you're given y(0) and v(0), you should be able to solve for k₁ and k₂ in the two exponential terms.

I'll take a look at my text tonight and brush up on the phase plane business. It's been 30 years since I looked at it last, so I'm pretty rusty on these ideas.

 

[Mark44]

Sorry to bother you again.

for y'' + 4y' + 13y = 0

The general solution would be y(t) = k₁e^-2tcost3t + k₂e^-2tsin3t + (27/145)cos2t + (24/145)sin2t.

It's a spiral sink, so would the phase plane (with inital conditions y(0) = 1, and v(0) = -2)look like this:

http://img13.imageshack.us/img13/8230/spiralsink.th.jpg [Broken]

not sure what to do since the eigenvectors have complex numbers.

and would the y(t) v(t) graphis look like this:

http://img13.imageshack.us/img13/4392/yvgraph.th.jpg [Broken]

According to my source, Differential Equations, Dynamical Systems, and Linear Algebra - Hirsch and Smale -- when you have two complex eigenvalues with negative real parts (your situation), you will have clockwise orbits spiraling into the origin.

Your phase plane (the first figure in post 9) should, I believe, show one trajectory starting at 1 on the y axis, and spiraling clockwise into the origin. Another trajectory should start at -2 on the v axis, I believe, and do the same. So your phase plane has more-or-less the right appearance, but is a little off in some details.

Your graph of y(t) and v(t) is probably OK, as both functions tend to zero as t increases, but y(t) starts at 1 (when t is 0) and v(t) starts at -2 (when t is 0).

Hope that's helpful.
Mark

 

[cse63146]

Your phase plane (the first figure in post 9) should, I believe, show one trajectory starting at 1 on the y axis, and spiraling clockwise into the origin. Another trajectory should start at -2 on the v axis

So there should be 2 spirals, one starting at y(0) = 1, and another spiral starting at v(0) = -2, and both going towards the origin?

 

[Mark44]

Right.

 

[cse63146]

would the phase portrait look the same if (for example) the original equation was y'' + 4y' + 13y = 3cost2t ?

 

[Mark44]

Let me get back to you. There is a fair amount of mental translation that I need to do, to convert your differential equations (the homogeneous one and the related nonhomogeneous one) into systems of first-order DEs, which is how things are presented in the text I cited.

Your 2nd order homogeneous DE is equivalent to a system of two 1st order DEs, which can be solved by matrix techniques. Your 2nd order nonhomogeneous DE is equivalent to a 4th order homogeneous DE, which is equivalent to a system of four 1st order DEs, which also can be solved by the use of matrices.

Your phase planes have axes marked as y and v (= dy/dx), while the ones I'm looking have axes of x₁ and x₂.

Going from a 2nd order DE to two 1st order DEs involves introducing a new variable v = y'. Going from a 2nd order nonhomogeneous DE to a 4th order homogeneous DE uses the concept of "annihilators" that I mentioned a while back in another thread.

Lurking in the background of all this are a lot of concepts from Linear Algebra, an area of mathematics that is inextricably intertwined with differential equations.

 

[cse63146]

If it helps, the general solution would be y(t) = k₁e^-2tcost3t + k₂e^-2tsin3t + (27/145)cos2t + (24/145)sin2t.

 

[Mark44]

Yes, I remember. You showed this several posts back.

 

[Mark44]

would the phase portrait look the same if (for example) the original equation was y'' + 4y' + 13y = 3cost2t ?

This is the question I said I'd come back to. The short answer is I don't think so. This nonhomogeneous 2nd order equation is equivalent to the homogeneous 4th order equation
(D² + 4D + 13)(D² + 4)y = 0

The characteristic equation is (r² + 4r + 13)(r² + 4) = 0, and the roots of this equation are r = -2 +/-3i, +/-2i. These are also the eigenvalues, and the corresponding eigenfunctions are e^{(-2 +/-3i)t} and e^(+/-2i)t. With the usual sleight of hand of taking suitable linear combinations to eliminate the imaginary parts, the eigenfunctions can be written as e^-2tsin3t, e^-2tcos3t, sin2t, and cos 2t.

Unlike the homogeneous 2nd order problem (which is equivalent to a system of two first-order DEs, and for which the solution space is two-dimensional), for the nonhomogeneous 2nd order system above, the solution space is four-dimensional. The eigenfunctions y_1,2 = e^{(-2 +/-3i)t} of the homogeneous DE span a two-dimensional space, and you can plot a graph of y₁ vs. y₂, which is nearly what you did in your phase portrait of y vs v (= y').

With the nonhomogeneous 2nd order problem (or the related homogeneous 4th problem), the eigenfunctions span a four-dimensional space, so right away there are problems in visualizing it. The highest dimension phase portrait shown in the text I've been looking at is three.

So anyway, based on a relatively quick perusal of my book, I don't believe that you can get a phase portrait for the nonhomogeneous equation you gave. It's possible that the authors are doing something to look at a projection of a higher dimension portrait down to a couple of dimensions, but I didn't catch that in my quick perusal.

 

[cse63146]

Thanks for all of your help. Just wrote my final about an hour ago (and I'm pretty sure I did pretty good; could be wrong). Would have done worse without your help, thanks again.

 

[Mark44]

Cool... Let me know how you did.