Direction of oscillations (2nd order ODE)

[rmiller70015]

Homework Statement

$$\frac{d\vec{Y}}{dt} = <br /> \begin{bmatrix}<br /> 0 & 2\\ <br /> -2 &-1 <br /> \end{bmatrix}\vec{Y}$$
With an initial condition of $$\vec{Y_0} = (-1,1)$$[/B]

a) Find the eigenvalues
b) Determine if the origin is a spiral sink, source, or center
c) Determine the natural period and frequency of the oscillations
d) Determine the direction of oscillations in the phase plane
e) Graph it

Homework Equations

The Attempt at a Solution

I have the solutions, the eigens are $$\frac{-1}{2} \pm \frac{\sqrt{15}i}{2}$$

Because λ is negative and the roots of the characteristic polynomial are complex the solutions oscillate and it is a sink.

The frequency and periods can be found with $$f = \frac{2π}{μ}$$

But the directions of oscillation is where I have a problem. Again I have the answer to this problem (graphing it with mathematica gives the directions), but my question is how do I typically find the direction of oscillations for equations of this form using algebra or some other relationship that exists?

Edit: I remember there being something about u + iw vectors, but it's really fuzzy that was the beginning of the semester.

 

[Ray Vickson]

Homework Statement

$$\frac{d\vec{Y}}{dt} = <br /> \begin{bmatrix}<br /> 0 & 2\\ <br /> -2 &-1 <br /> \end{bmatrix}$$
With an initial condition of $$\vec{Y_0} = (-1,1)$$[/B]

a) Find the eigenvalues
b) Determine if the origin is a spiral sink, source, or center
c) Determine the natural period and frequency of the oscillations
d) Determine the direction of oscillations in the phase plane
e) Graph it

Homework Equations

The Attempt at a Solution

I have the solutions, the eigens are $$\frac{-1}{2} \pm \frac{\sqrt{15}i}{2}$$

Because λ is negative and the roots of the characteristic polynomial are complex the solutions oscillate and it is a sink.

The frequency and periods can be found with $$f = \frac{2π}{μ}$$

But the directions of oscillation is where I have a problem. Again I have the answer to this problem (graphing it with mathematica gives the directions), but my question is how do I typically find the direction of oscillations for equations of this form using algebra or some other relationship that exists?

Edit: I remember there being something about u + iw vectors, but it's really fuzzy that was the beginning of the semester.

Your DE makes no sense: it equates a vector on one side to a matrix on the other side. Did you mean
$$\frac{d}{dt} \vec{Y}(t) = \begin{bmatrix}<br /> 0 & 2\\<br /> -2 &-1<br /> \end{bmatrix} \vec{Y}(t) \: ?$$

 

[rmiller70015]

Your DE makes no sense: it equates a vector on one side to a matrix on the other side. Did you mean
$$\frac{d}{dt} \vec{Y}(t) = \begin{bmatrix}<br /> 0 & 2\\<br /> -2 &-1<br /> \end{bmatrix} \vec{Y}(t) \: ?$$

Yes I did, sorry about that forgot my second vector.

 

[rmiller70015]

So, I figure it out eventually and just in case any futuristic internet people find this I figure I can be helpful by saying if you use the point (1,0) and plug it into the original equation and set it equal to the initial condition vector you can draw an arc from the point (1,0) to the next point and consider if the solution is a source or sink.