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
x' = 0 = c( y + x - x3/3 ) (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 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 = a - x + bx - b/3x3 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 - bx2 Partial f Partial x
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.x = +- sqrt( b - 1 / b )
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.
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.
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,
c( 1 - x2) 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( c2 - b / c )
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.a = +- ( b/c sqrt( c2 - b ) - 1/c sqrt( c2 - b ) - 1/3 ( b( c2 - b )3/2) / c 3 )