8.3: Two-Dimensional Bifurcations

a Hopf bifurcation. The Hopf bifurcation comes in two types: supercritical Hopf bifurcation and subcritical Hopf bifurcation. For the supercritical Hopf bifurcation, as \(\mu\) increases slightly above zero, the resulting oscillation around the now unstable fixed point is quickly stabilized at small amplitude. This stable orbit is called a limit cycle. For the subcritical Hopf bifurcation, as \(\mu\) increases slightly above zero, the limit cycle immediately jumps to large amplitude.

Supercritical Hopf Bifurcation

A simple example of a supercritical Hopf bifurcation can be given in polar coordinates:

\[\begin{aligned}\overset{.}{r}&=\mu r-r^3 , \\ \overset{.}{\theta}&=\omega +br^2,\end{aligned}\] where \(x = r \cos\theta\) and \(y = r \sin\theta\). The parameter \(\mu\) controls the stability of the fixed point at the origin, the parameter \(\omega\) is the frequency of oscillation near the origin, and the parameter \(b\) determines the dependence of the oscillation frequency at larger amplitude oscillations. Although we include \(b\) for generality, our qualitative analysis of these equations will be independent of \(b\).

The equation for the radius is of the form of the supercritical pitchfork bifurcation. The fixed points are \(r_* = 0\) and \(r_* =\sqrt{\mu}\) (note that \(r > 0\)), and the former fixed point is stable for \(\mu < 0\) and the latter is stable for \(\mu > 0\). The transition of the eigenvalues of the Jacobian from negative real part to positive real part can be seen if we transform these equations to cartesian coordinates. We have using \(r^2 = x^2 + y^2\), \[\begin{aligned}\overset{.}{x}&=\overset{.}{r}\cos\theta -\overset{.}{\theta}r\sin\theta \\ &=(\mu r-r^3)\cos\theta -(\omega +br^2)r\sin\theta \\ &=\mu x-(x^2+y^2)x-\omega y-b(x^2+y^2)y \\ &=\mu x-\omega y-(x^2+y^2)(x+by); \\ \overset{.}{y}&=\overset{.}{r}\sin\theta +\overset{.}{\theta}r\cos\theta \\ &=(\mu r-r^3)\sin\theta +(\omega +br^2)r\cos\theta \\ &=\mu y-(x^2+y^2)y+\omega x+b(x^2+y^2)x \\ &=\omega x+\mu y-(x^2+y^2)(y-bx).\end{aligned}\]

The stability of the origin is determined by the Jacobian matrix evaluated at the origin. The nonlinear terms in the equation vanish and the Jacobian matrix at the origin is given by \[J=\left(\begin{array}{cc}\mu &-\omega \\ \omega &\mu\end{array}\right).\nonumber\]

The eigenvalues are the solutions of \((\mu −\lambda)^2 + \omega^2 = 0\), or \(\lambda = \mu ± i\omega\). As \(\mu\) increases from negative to positive values, exponentially damped oscillations change into exponentially growing oscillations. The nonlinear terms in the equations stabilize the growing oscillations into a limit cycle.

Subcritical Hopf Bifurcation

The analogous example of a subcritical Hopf bifurcation is given by \[\begin{aligned}\overset{.}{r}&=\mu r+r^3-r^5, \\ \overset{.}{\theta}&=\omega +br^2.\end{aligned}\]

Here, the equation for the radius is of the form of the subcritical pitchfork bifurcation. As \(\mu\) increases from negative to positive values, the origin becomes unstable and exponentially growing oscillations increase until the radius reaches a stable fixed point far away from the origin. In practice, it may be difficult to tell analytically if a Hopf bifurcation is supercritical or subcritical from the equations of motion. Computational solution, however, can quickly distinguish between the two types.