Analysis

Limit Point Bifurcation

In our analysis of the Fitzhugh-Nagumo model we first noted that we couldn't solve the system of equations given by setting x' and y' equal to zero.
x' = 0 = c( y + x - x3/3 ) (1)
y' = 0 = -(( x - a + by) /c ) (2)
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 y
y = a - x / b
and then substituted this into equation (1). This gave us the following equation for the fixed points of the system.
0 = a - x + bx - b/3x3 SS equation
This gives us an equation for the fixed points of the system in terms of a, b and x. Next since this equation that represents the fixed points of the system is of only a single variable we can take the derivative of this equation and it will tell us about the stability of the system at it's fixed points.
0 = - 1 + b - bx2 Partial f Partial x
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.
x = +- sqrt( b - 1 / b )
And since these points are only a function of the value of b. We can substitute this equation above that tells us when the fixed points change characteristics ( a.k.a. non-hyperbolic fixed points) into the steady state equation. What we get is an equation in terms of the parameters a and b that tells us for what values of a and b we will have non-hyperbolic fixed points. The limit point bifurcation.
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.
This equation creates a line that cuts the a b plane into two regions of differing characteristics. Where the upper region has three fixed points, one saddle and two stable nodes, and the lower region only has one stable node.

The limit point bifurcation is the white line shown here in the a b plane. The other line will be explained later.

Hopf Bifurcations

Since the manipulations involved in getting the solutions ( limit point bifurcations ) above eliminated c from our equations we knew our analysis couldn't be complete. So we next looked at the Jacobian for the system above.
c( 1 - x2) c
-1/c -b/c
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. (Hopf Bifurcations) We found these by finding the trace of the Jacobian and seting it equal to zero,
x = +- sqrt( c2 - 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.
a = +- ( b/c sqrt( c2 - b ) - 1/c sqrt( c2 - b ) - 1/3 ( b( c2 - b )3/2) / c 3 )
We arrive at two equations that give us another set of lines in the a b plane who's positions depend on the value of c. Specificaly 0 < c <= 1 gives you one more region below the limit point bifurcation line that contains limit cycles and one node. And for values of c > 1 you get six regions with differing characteristics that will be shown below.

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 c, we must choose a value of c and then construct a bifurcation diagram of a vs. b. In order to analyze the two cases (0 < c <= 1 and c > 1) we chose a value of c for each case. Specifically, for case 1 c = 1, and for caes 2 c = 2. For c=1 the hopf bifurcation is represented by the red line, (recall the white line represents limit point bifurcation). For c=2 the hopf bifurcation is represented by both the red and green lines, (once again the white line represents limit point bifurcation. Click here for a closer look atregions B and C of the bifurcation diagram for c=2. As you will see below, the red (lower hopf bifurcation) curve determines the region(s) in which there are periodic (limit cycles) solutions to the system of differential equations for any value of c.

Phaseportraits

The following three phaseportraits demonstrate the limit point and hopf bifurcation analysis for case 1:
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