4.6 Hopf bifurcations

We discussed, in the last section, various types of bifurcations that occur at an equilibrium point $\ol{x}$ of the system

$$x' = f(x, \mu )$$ (4.38)

when the matrix $Df(\ol{x}, \mu_{0})$ has a simple zero eigenvalue. In this section, we consider bifurcations that occur when the matrix $Df(\ol{x}, \mu_{0})$ has a simple pair of pure imaginary eigenvalues and no other eigenvalues with zero real part. In this case, a Hopf bifurcation can occur and a periodic orbit is created as $\mu$ passes through the bifurcation value $\mu_{0}$. Let us first consider a simple example.

Example 4.6.1:     Consider the two-dimensional system

$$\left\{ \begin{array}{l} x_{1}' = x_{2} + x_{1} (\mu - x_{1}^... ...+ x_{2} (\mu - x_{1}^{2} - x_{2}^{2}) \, , \end{array} \right.$$ (4.39)

where $\mu \in R$ is a parameter.

There is only one equilibrium point (0,0) and

$$Df (0, \mu ) = \left[ \begin{array}{rc} \mu & 1 \\ -1 & \mu \end{array} \right] \, .$$

Thus the origin is asymptotically stable if $\mu < 0$ and unstable if $\mu > 0$. For $\mu = 0$, Df (0,0) has a pair of pure imaginary eigenvalues. The structure of the phase portrait becomes apparent if we rewrite system (4.39) in polar coordinates. Let $x_{1} = \rho \cos \theta$ and $x_{2} = \rho \sin \theta$. Then (4.39) reduces to

$$\left\{ \begin{array}{l} \rho' = \rho (\mu - \rho^{2}) \, , \\ \theta' = - 1 \, . \end{array} \right.$$ (4.40)

We see from (4.40) that when $\mu \leq 0$, the origin is asymptotically stable and when $\mu > 0$, the origin becomes unstable and there is a periodic orbit $\gamma_{\mu}$ given by

$$\gamma_{\mu} : \left\{ \begin{array}{l} x_{1} (t) = \sqrt{\mu... ..., , \\ x_{2} (t) = \sqrt{\mu} \cos t \, , \end{array} \right.$$ (4.41)

which is asymptotically stable. Thus $\mu = 0$ is a bifurcation value. The phase portraits and bifurcation diagram are shown below.

Figure 4.6.1: Phase portraits for system (4.39)

Figure 4.6.2: The bifurcation diagram

This type of bifurcation is called a Hopf bifurcation.

Next, we consider system (4.38) in R² and assume that it be written into the form

$$\left\{ \begin{array}{l} x_{1}' = \mu x_{1} - x_{2} + p (x_{1... ...2}' = x_{1} + \mu x_{2} + q (x_{1}, x_{2}), \end{array}\right.$$ (4.42)

where

$$\begin{eqnarray*}p (x_{1}, x_{2}) & = & \sum_{2 \leq i + j \leq 3} a_{ij} x_{1}^... ... 0 (x_{1}^{\alpha} x_{2}^{\beta}), \quad \alpha + \beta = 3 \, . \end{eqnarray*}$$

Let

$$\sigma = \frac{3 \pi}{2} [3(a_{30} + b_{03}) + (a_{12} + b_{2... ...02}) + a_{11}(a_{02}+ a_{20}) - b_{11} (b_{02} + b_{20})] \, .$$ (4.43)

Then we have the following theorem (L. Perko, p. 317).

Theorem 4.6.1: (Hopf Bifurcation)     If $\sigma \neq 0$, then a Hopf bifurcation occurs at the origin of system (4.42) at the bifurcation value $\mu = 0$; in particular, if $\sigma < 0$, then system (4.42) has a one-parameter family of stable limit cycles for $\mu > 0$ and no limit cycles for $\mu \leq 0$, and if $\sigma > 0$, then system (4.42) has a one-parameter family of unstable limit cycles for $\mu < 0$ and no limit cycles for $\mu \geq 0$.