finding the direction of the Nullcline?

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Nov 22nd 2009, 07:21 PM
bsanghera

finding the direction of the Nullcline?

I need help finding the direction the arrows point when graphing the phase portrait two system of equations. What equation do i plug in points to find out which way the arrows point? dx/dt dy/dt or is there another equation I'm supposed to use which I'm forgetting?
Jan 19th 2010, 04:25 AM
Rebesques

Quote:

Originally Posted by bsanghera

I need help finding the direction the arrows point when graphing the phase portrait two system of equations. What equation do i plug in points to find out which way the arrows point? dx/dt dy/dt or is there another equation I'm supposed to use which I'm forgetting?

It would be best if you compute the right hand sides over a grid of points (x,y).
Jan 19th 2010, 05:43 AM
shawsend

Quote:

Originally Posted by bsanghera

I need help finding the direction the arrows point when graphing the phase portrait two system of equations. What equation do i plug in points to find out which way the arrows point? dx/dt dy/dt or is there another equation I'm supposed to use which I'm forgetting?

The system:

$\frac{dx}{dt}=f(x,y)$

$\frac{dy}{dt}=g(x,y)$

can be considered as a vector-valued function in the x-y plane and I can write it as:

$\textbf{V}\left(\begin{array}{c}x \\ y\end{array}\right)=\left(\begin{array}{c} f(x,y) \\ g(x,y)\end{array}\right)$

so at the point (2,1) I have a vector:

$\textbf{V}\left(\begin{array}{c} 2 \\ 1 \end{array}\right)=\left(\begin{array}{c}f(2,1) \\ g(2,1)\end{array}\right)$

and it's not too hard to plot these vectors at each point in the x-y plane.

However, the newest version of Mathematica, ver. 7 has a function StreamPlot to do this automatically. See this thread where I gave an example:

http://www.mathhelpforum.com/math-he...ium-level.html

and I recommend "Differential Equations" by Blanchard, Devaney, and Hall. It's an easy read. :)
Jan 20th 2010, 05:42 AM
HallsofIvy

The sign of either dx/dt or dy/dt should tell you that. If dx/dt> 0, the phase portait arrows is to the right, if dx/dt< 0, to the left. Similarly, if dy/dt> 0, they are upward, if dy/dt< 0, they are downward. Notice that if dy/dt> 0 and dx/dt> 0, then dy/dx> 0 so "right" and "up" are the same.

By the way, you titled this "finding the direction of the Nullcline?". The Nullclines are NOT arrows in the phase portrait. They are lines or curves (passing through points where dx/dt= dy/dt= 0), not "arrows" or vectors, and do not have a "direction".