Hopf bifurcation

In order to study the dynamics near the critical point we will derive the center manifold equations using the technique discussed earlier in the course. Assume that $\alpha$ is fixed; then the critical elasticity number is determined by the Deborah number ${\rm De}$. Thus as ${\rm De}$ increases past a critical value which we denote ${\rm De}_{c}$ a Hopf bifurcation occurs. Let

$${\rm De}={\rm De}_{c}+\lambda$$

so that the bifurcation parameter is $\lambda$. The Birkhoff normal form for the equation on the center manifold is  

$$\frac{\partial z}{\partial t}=\sigma(\lambda) z+b(\lambda)z\vert z\vert^{2}$$ (207)

and the center manifold has the form  

$$y=\psi(\lambda)z^{2}+\bar{\psi}(\lambda)\bar{z}^{2}+\chi(\lambda)z\bar{z}$$ (208)

Here $\sigma=i\omega_{c}+\lambda a$. The leading terms of the coefficients for selected values of the parameters are given in table 6.1. The bifurcations are of the pitchfork type. The direction of the bifurcation as well as the stability of the bifurcating solutions are determined by the real part of b. If $\Re(b)<0$, the bifurcation is supercritical and stable but if $\Re(b)\gt$ it is subcritical and unstable. From table 6.1, we see that the bifurcation is supercritical and stable if $\beta=1$ and subcritical and unstable if $\beta= 1/2$. In fact, there is a value $\beta_{c}$ of $\beta$ such that the bifurcation is supercritical and stable if $\beta\gt \beta_{c}$ and subcritical if $\beta <\beta_{c}$. The value of $\beta_{c}$ lies in the interval (0.97, 0.98). This result agrees with experimental results of McKinley et al [2]. In experiments with a Boger fluid for which $\beta = 0.41$ they observed a subcritical Hopf bifurcation.

   

$$\beta \omega_{c}$$ De_c a b

1.00 1.45531 1.0 0.68714 + 0.68714i -0.11203 + 1.49435i

0.50 1.45531 1.0 0.68714 + 0.68714i 0.03149 + 5.05326i