Transcritical bifurcation

The normal or generic form of this bifurcation is given by

.................................................................................................(4.2.13)

So, in this case the equilibrium point is obtained by substituting and they are x=0 (trivial state)and (non trivial state). Here, the eigenvalue can be given by . Figure 4.2.5 shows the variation of x with .

Here for branch AO, λ is negative and hence this branch is stable.

for Branch OB, λ is positive, hence it is unstable.

for Branch CO, , (positive), So this branch is unstable

for Branch OD, λ is negative and hence this branch is stable.

It may be observed that the trivial and non-trivial branches change their stability at the origin and so the name of this bifurcation point is transcritical bifurcation.

Figure 4.2.5: Variation of x with showing transcritical bifurcation at point O.

Hopf bifurcation: For a system , following two conditions to be satisfied at the dynamic bifurcation point (Marsden and McCracken 1976, Nayfeh and Balachandran 1995).

1.