This prompted us to use a different approach for finding the fixed points of the non-linear system. First we set both equations equal to zero as before, but this time we solved equation (2) for

x' = 0 = c( y + x - x ^{3}/3 )(1) y' = 0 = -(( x - a + by) /c ) (2)

and then substituted this into equation (1). This gave us the following equation for the fixed points of the system.y = a - x / b

This gives us an equation for the fixed points of the system in terms of0 = a - x + bx - b/3x^{3}SS equation

But we're looking for a bigger picture. Our goal is to understand as much as possible about the characteristics of the system of differential equations in terms of its parameters as we can. So to start we note that if we find the points where the derivative is equal to zero, then we'll know at what points the characteristics of the system will change. Specificaly the eigenvalues change from negative to positive and vice-versa.0 = - 1 + b - bx^{2}Partial f Partial x

And since these points are only a function of the value ofx = +- sqrt( b - 1 / b )

This equation creates a line that cuts the

a = sqrt( (b - 1) / b)b - sqrt( (b - 1)/ b ) - b/3( (b - 1)/b) ^{3/2}Note: Only the negative squareroot gave us positive values for the parameters.

The limit point bifurcation is the white line shown here in the

And since we already found all of the non-hyperbolic fixed points that correspond to zero eigenvalues. We focus our attention on the non-hyperbolic fixed points corresponding to pure imaginary eigenvalues.

c( 1 - x ^{2})c -1/c -b/c

And we also made sure that the determinant was greater than zero this guarantees that the eigenvalues will be imaginary, and the eigenvalues will have a zero real part. Then by plugging the resulting equations from above back into the equation for the fixed points.x = +- sqrt( c^{2}- b / c )

We arrive at two equations that give us another set of lines in thea = +- ( b/c sqrt( c^{2}- b ) - 1/c sqrt( c^{2}- b ) - 1/3 ( b( c^{2}- b )^{3/2}) / c^{3})

Now with the equations found above, it is possible to create a bifurcation set (parameter vs. parameter) for the original system of differential equations that describes the stability (characteristics) of the fixed points in terms of the parameters of the differential equations. Since the hopf bifurcation depends on

Phaseportrait for Case 1 region A Parameters for region A are below both the limit point and hopf bifurcation curves, so solutions for these parameter sets have one unstable fixed point and a limt cycle.

Phaseportrait for Case 1 region B Parameters for this region are below the limit point bifurcation, but above the hopf bifurcation so solutions for these parameter sets have one stable fixed point.

Phaseportrait for Case 1 region C Parameters for this region are above both the limit point and hopf bifurcation curves, so solutions for these parameter sets have one saddle and two stable fixed points.

The following six phaseportraits demonstrate the limit point and hopf bifurcation analysis for case 2:

Phaseportrait for Case 2 region A Parameters for this region are below the limit point bifurcation and both hopf bifurcation curves, so solutions for these parameter sets have one unstable fixed point and a limit cycle.

Phaseportrait for Case 2 region B Parameters for this region are above the limit point bifurcation, but below the both hoph bifurcation curves so solutions for this parameter range contain two unstable fixed points and a saddle point within a limit cycle.

Phaseportrait for Case 2 region C Parameters for this region are above the limit point bifurcation, above the lower hopf bifurcation, and below the the upper hopf bifurcation, so solutions for this parameter range have three fixed points. There is one unstable, one stable and one saddle fixed point.

Phaseportrait for Case 1 region D Parameters for this region fall under the limit point bifurcation. There is one stable fixed point.

Phaseportrait for Case 1 region E Parameters for this region are above the limit point and lower hopf bifurcation curves, but within the upper bifurcation 'loop.' Solutions with this range of parameter values have two stable fixed points and a saddle point.

Phaseportrait for Case 1 region F Parameters for this region are above the limit point and lower hopf bifurcation curves, and are not within the upper bifurcation 'loop.' Solutions with this range of parameter values have two stable fixed points and a saddle point.

[ Introduction | Background | Conclusions ]

Mike Willock willock@wpi.wpi.edu

Richard Demar rdemar@wpi.wpi.edu