In part 4 of the first project we were asked to find the bifurcation value for
and describe the phase portraits for 0< b<and b>.
To obtain the bifurcation value we have to calculate the Jacobean matrix and evaluate it at the equilibrium points.
The Jacobeanmatrix is
And there are three equilibrium points (0,0) (1.16,0) (-1.16,0)
Evaluating at (0,0) gives the matrix:
J (0,0) = with characteristicpolynomial . When analyzing the polynomial: we can see that because of the signs, no matter the value of b, the origin is always going to bea saddle. Therefore the bifurcation value does not depend on this equilibrium point because the damping coefficient does not have theability to change the phase portrait.
Because the only x term in the Jacobean matrix is squared the analysis for the equilibrium points (1.16,0)(-1.16,0) is the same.
J(1.16,0) = with characteristic polynomial .
The equation that determines the Eigen values is. If has complexnumbers the equilibrium point is a spiral sink, and if only has real number the equilibrium point is a sink. Therefore the value of b where thesquared root is zero is the bifurcation value:
We can then conclude that the equilibrium points (1.16,0) are the ones that separate thedifferent possible behaviors of the beam because are the only ones that we can change fro spiral sink to sink by changing the bifurcation value.