Andronov-Hopf bifurcation theorem

Assume that, near $\mu=\mu^*$, the Jacobian matrix of the vector field evaluated at $(\bar{x}(\mu),\bar{y}(\mu))$ has eigenvalues of the form

$$ a(\mu)\pm i b(\mu) $$


$$ a(\mu^*)=0\quad\text{and}\quad b(\mu^*)\neq 0. $$

Also assume that the real parts of the eigenvalues change signs as $\mu$ is varied through $\mu^*$, i.e.,

$$ \frac{\text{d}a}{\text{d}\mu}(\mu^*)\neq 0. $$

Given these hypotheses, the following possibilities arise :

  • There is a range $\mu^*<\mu<\mu_1$ such that a limit cycle surrounds $(\bar{x}(\mu),\bar{y}(\mu))$. As $\mu$ is varied, the diameter of the limit cycle changes in proportion to $\sqrt{|\mu-\mu^*|}$. There is no other closed trajectory near $(\bar{x}(\mu),\bar{y}(\mu))$. This case is termed a supercritical bifurcation.
  • There is a range $\mu_0<\mu<\mu^*$ such that a similar conclusion to the previous case holds. This case is termed a subcritical bifurcation.