Consider the differential equation dxdtkxx3 depending on the

Consider the differential equation

dxdt=kx?x3

depending on the parameter k .

Determine the bifurcation point for the parameter k .
k=

If k is greater than the bifurcation point, then there Chooseis one stable critical pointis one semistable critical pointis one unstable critical pointare two critical points, both of which are stableare two critical points, both of which are unstableare two critical points, both of which are semistableare two critical points, one of which is unstable and one of which is stableare two critical points, one of which is unstable and one of which is semistableare two critical points, one of which is stable and one of which is semistableare three critical points, all of which are stableare three critical points, all of which are semistableare three critical points, all of which are unstableare three critical points, one of which is stable, the other two of which are semistableare three critical points, one of which is stable, the other two of which are unstableare three critical points, one of which is semistable, the other two of which are stableare three critical points, one of which is semistable, the other two of which are unstableare three critical points, one of which is unstable, the other two of which are stableare three critical points, one of which is unstable, the other two of which are semistableare three critical points: one stable, one semistable and one unstableare no critical points .

If k is equal to the bifurcation point, then there Chooseis one stable critical pointis one semistable critical pointis one unstable critical pointare two critical points, both of which are stableare two critical points, both of which are unstableare two critical points, both of which are semistableare two critical points, one of which is unstable and one of which is stableare two critical points, one of which is unstable and one of which is semistableare two critical points, one of which is stable and one of which is semistableare three critical points, all of which are stableare three critical points, all of which are semistableare three critical points, all of which are unstableare three critical points, one of which is stable, the other two of which are semistableare three critical points, one of which is stable, the other two of which are unstableare three critical points, one of which is semistable, the other two of which are stableare three critical points, one of which is semistable, the other two of which are unstableare three critical points, one of which is unstable, the other two of which are stableare three critical points, one of which is unstable, the other two of which are semistableare three critical points: one stable, one semistable and one unstableare no critical points .

If k is less than the bifurcation point, then there Chooseis one stable critical pointis one semistable critical pointis one unstable critical pointare two critical points, both of which are stableare two critical points, both of which are unstableare two critical points, both of which are semistableare two critical points, one of which is unstable and one of which is stableare two critical points, one of which is unstable and one of which is semistableare two critical points, one of which is stable and one of which is semistableare three critical points, all of which are stableare three critical points, all of which are semistableare three critical points, all of which are unstableare three critical points, one of which is stable, the other two of which are semistableare three critical points, one of which is stable, the other two of which are unstableare three critical points, one of which is semistable, the other two of which are stableare three critical points, one of which is semistable, the other two of which are unstableare three critical points, one of which is unstable, the other two of which are stableare three critical points, one of which is unstable, the other two of which are semistableare three critical points: one stable, one semistable and one unstableare no critical points

Solution

We will get the burification points by equation the dx/dt term equal to zero

dx/dt = 0 => kx - x^3

kx - x^3 = 0

k = x^2

Now depending on the value of it will be always positive

Hence stable region will be for any x>0

semi-stable state will occur when x=0 making it not stable and not unstable at the same moment