Understanding Phase Curves and Directionality in ODE Systems

[MathematicalPhysicist]

Well I need to find the phase graph of the next system of ode:
dx2/dt=-4x1
dx1/dt=x2
now i know the curves of x2 as a function of x1 are ellipses, but in what direction.
I mean obviously i need to check dx2/dx1, and from this find if x2 is decreasing or increasing, so for the first quadrant obviously it goes clockwise cause dx2/dx1<0 so x2 in this quadrant is decreasing, the same analysis i did with the other quadrants but shouldn't it have a consistent direction, I mean from my analysis not every part of the ellipse in every quadrant moves clockwise.

can this be ok?
or am i way off here?

thanks in advance.

 

[HallsofIvy]

What happens at the point (1, 0)? If x1= 1 and x2= 0, then dx2/dt= -4 and dx1/dt= 0: x2 is decreasing while x1 stays the same. That's going downward and indicates that the flow around the ellipse is clockwise.

Your statement "from my analysis not every part of the ellipse in every quadrant moves clockwise" is just wrong. if x1= 0 and x2= 1, dx2/dt= 0, dx1/dt= 1 so the flow is to the right: again clockwise. If x1= 0 and x2= -1, dx2/dt= 0, dx1/dt= -1: clockwise. If x1= -1 and x2= 0, dx2/dt= 4, dx1/dt= 0: clockwise.

 

[MathematicalPhysicist]

my way is like this:
dx2/dx1=-(4x1/x2) so for x1>0 and x2<0 dx2/dx1>0, so x2 should increase in this quadrant, should it not?

I hope you can clear this issue to me.
thanks in advance.

 

[HallsofIvy]

my way is like this:
dx2/dx1=-(4x1/x2) so for x1>0 and x2<0 dx2/dx1>0, so x2 should increase in this quadrant, should it not?

I hope you can clear this issue to me.
thanks in advance.

As x1 increases, yes x2 increases: the tangent line to the ellipse is increasing.
And as x1 decreases, x2 decreases.

But that's not the question! You are talking about what happens as t increases. In the fourth quadrant, both x1 and x2 decrease as t increases.

The flow around the ellipse is clockwise.

 

[MathematicalPhysicist]

ok, thanks.

 

[MathematicalPhysicist]

i have another question:
how can i tell when it's counterclockwise or clockwise telling from those derivatives?
I mean in clockwise the angle which is also the slope is defined to be negative, i feel that i read it in courant's first volume of intro to clalc, but can't seem to remember.