Describing anEquilibrium Solution which is neither Stable or unstable?

[raaznar]

Homework Statement

Discuss the stability of the equilibrium solutions

Homework Equations

dy/dx = cos^2(y) between 0<=y<=2pi
y(0)=0
y(0)=pi/2
y(0)=pi

The Attempt at a Solution

Found the equilibrium solutions to be pi/2 and 3pi/2.
Rough graphed y(x) which was a y vs x graph with horizontal lines (Eq Solu) at pi/2 and 3pi/2.
Then added the curves y(0)=0 and y(0)=pi which are graphs approaching Eq Sols pi and 3pi/2 respectively.

Now, I'm not sure how to describe the equilibrium solutions. The graph is neither stable or unstable but just keeps repeating in the given domain. How would I describe this type of behaviour? Thanks

 

[Dick]

Homework Statement

Discuss the stability of the equilibrium solutions

Homework Equations

dy/dx = cos^2(y) between 0<=y<=2pi
y(0)=0
y(0)=pi/2
y(0)=pi

The Attempt at a Solution

Found the equilibrium solutions to be pi/2 and 3pi/2.
Rough graphed y(x) which was a y vs x graph with horizontal lines (Eq Solu) at pi/2 and 3pi/2.
Then added the curves y(0)=0 and y(0)=pi which are graphs approaching Eq Sols pi and 3pi/2 respectively.

Now, I'm not sure how to describe the equilibrium solutions. The graph is neither stable or unstable but just keeps repeating in the given domain. How would I describe this type of behaviour? Thanks

Repeating is not the important feature, it's whether a solution starting near your equilibrium solution will flow into or out of the equilibrium solution. If you start with an initial condition a little below y=pi/2, what does it do? What about a little above?

 

[HallsofIvy]

If, as you appear to be saying, the solutions near the equilibrium solution tend neither toward it nor away from it but circulate around it, then the equilibrium solution is a "center": there are periodic solutions in its vicinity.