Part II: Two-dimensional systems: flows in the plane - Chapter 8


Here we extend what we have seen on bifurcations in one-dimensional systems. In dimension two, we still find that fixed points can be created or destroyed or destabilized as we vary a parameter. But there is a novelty: periodic solutions are possible, and there are ways to turn them on or off by tuning a parameter. This is the so-called the Poincaré-Andronov-Hopf bifurcation.
Let us stress that we just open the door of a broad field about which entire books are written.

Saddle-node bifurcation & a pendulum with torque

This bifurcation is the basic mechanism for the creation and destruction of fixed points. The following system is the prototypical example:

$$ \begin{cases} \dot{x}= \mu-x^2\\ \dot{y}=-y \end{cases} $$

where $\mu\in [-1,1]$. We have uncoupled equations that are easy to analyze. The first equation is nothing but the example of saddle-node bifurcation we’ve seen in dimension one. The second equation describes exponentially damped solutions in the $y$-direction. In other words, this bifurcation is a fundamentally one-dimensional event.

Example. We come back to the damped pendulum. We add a new feature: a constant torque $\mu>0$ is applied to the system (concretely, imagine air blown through a straw that pushes on the vanes of a pinwheel fixed at the rod, at the level of the pivot). This gives the equations

$$ \begin{cases} \dot{x}= y\\ \dot{y}=\mu-\sin x -\lambda y. \end{cases} $$

Transcritical bifurcation

In this bifurcation, there is an exchange of stability between two fixed points. The canonical equations for this are

$$ \begin{cases} \dot{x}= \mu x-x^2\\ \dot{y}=-y \end{cases} $$

where $\mu\in [-1,1]$. Again, this system is constructed from the one-dimensional case by adding an exponentially damped motion in the $y$-direction.

Pitchfork bifurcation & a bob in a rotating hoop

In this bifurcation, a fixed point splits into two other fixed points as a parameter passes a critical value. Here is a prototypical example:

$$ \begin{cases} \dot{x}= \mu x-x^3\\ \dot{y}=-y \end{cases} $$

where $\mu\in [-1,1]$.

This bifurcation is said to ‘supercritical’ because the appearance of the symmetric fixed points arises for $\mu$ above the critical value $\mu=0$. We now look at the subcritical pitchfork bifurcation, which is somewhat the ‘inverted version’ of the supercritical one. This time, we go from a situation where there is a unique unstable fixed point to a situation where it becomes stable and is accompanied by two unstable fixed points.

$$ \begin{cases} \dot{x}= \mu x+x^3\\ \dot{y}=-y \end{cases} $$

where $\mu\in [-1,1]$.

Example: a bob in a rotating hoop. Consider a bob of mass $m$ constrained on a circular hoop. Gravitational and frictional forces, as well as constraint forces that keep is on the hoop, are acting on it. The hoop itself is spun about a vertical axis with constant angular velocity $\omega$.

Using Newton’s law, it can be shown that

$$ \ddot{\theta}=\omega^2 \cos\theta \sin\theta-\omega_0^2\sin\theta -\frac{\lambda}{m} \dot{\theta} $$

where $g$ is the acceleration of gravity and $\omega_0=\sqrt{\frac{g}{\ell}}$. We take into account friction effects by the term $-\lambda \dot{\theta}$, where $\lambda\geq 0$. The case $\lambda=0$ corresponds to the absence of friction. We can choose units in such a way that $m=g=\ell=1$, whence $\omega_0=1$.

Setting $x=\theta$ and $y=\dot{\theta}$ gives the planar system

$$ \begin{cases} \dot{x}= y\\ \dot{y}= \omega^2 \sin x \cos x- \sin x -\lambda y. \end{cases} $$

Observe that when $\omega=0$, we recover the equations of the basic pendulum. Hence

We are going to observe that the value $\omega_c=1$ is the critical rotation rate above which the bob rises, though we start from the rest position. In the language of bifurcations, the system undergoes a supercritical pitchfork bifurcation. When $\omega<1$, there are only two fixed points, namely $(0,0)$ and $(\pi,0)$. When $\omega>1$, there four fixed points, namely $(0,0)$, $(\pi,0)$, $(-\arccos\big(\frac{1}{\omega^2}\big),0)$ and $(\arccos\big(\frac{1}{\omega^2}\big),0)$. Linearizing about $(\pi,0)$, one can check that we have a saddle for all values of $\omega$. This fixed point plays no role in the bifurcation.

Creating & destroying limit cycles, & Hopf’s bifurcation theorem

We present a diagnostic tool that can be used in establishing the existence of a limit cycle, namely Hopf’s bifurcation theorem. In a Hopf (or PoincarĂ©-Andronov-Hopf) bifurcation, a source becomes a sink, or vice-versa, as a pair of complex conjugate eigenvalues of the linearization at the fixed point cross the imaginary axis of the complex plane. In other words, there is a critical value for which the linearized vector field has zero trace and positive determinant. Under reasonably generic assumptions about the system, one can expect to see a small-amplitude limit cycle branching from the fixed point.

Consider the system

$$ \begin{cases} \dot{x}= \mu x-y -x(x^2+y^2)\\ \dot{y}=x+\mu y - y(x^2+y^2) \end{cases} $$

where $\mu$ is a parameter.

Let’s analyse this system. We start by linearizing the system at $(0,0)$, which is the only fixed point. This amounts to keeping only the linear terms:

$$ \begin{pmatrix} \dot{x}\\ \dot{y} \end{pmatrix} = \begin{pmatrix} \mu & -1\\ 1 & \mu \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}. $$

The determinant of the matrix is $1+\mu^2$, which is always $>0$, and its trace is $2\mu$. The eigenvalues are

$$ \lambda_{\pm}=\mu\pm i. $$

Therefore, as $\mu$ increases through zero, the eigenvalues cross the imaginary axis, and the origin shifts from being a spiral sink to being a spiral source.
So, we understood locally around the origin the behavior of solutions. Shifting to polar coordinates ($x=r\cos\theta$, $y=r \sin\theta$), we can complete the picture. Indeed, we get the decoupled equations

$$ \begin{cases} \dot{r}=r(\mu-r^2)\\ \dot{\theta}=1. \end{cases} $$

The second equation tells us that solutions rotate counterclockwise at constant angular velocity. The first one is easy to understand as well. for $\mu<0$, all solutions tend towards the origin. For $\mu>0$, solutions spiral inwards for $r>\sqrt{\mu}$ and outward for $r<\sqrt{\mu}$. The circle of radius $r=\sqrt{\mu}$ is of course the limit cycle. We leave the reader to cook up a ‘clockwise rotating’ version of the system.

The system we’ve just studied is in some sense the prototypical example of a supercritical Hopf bifurcation. It also comes in a ‘subcritical version’:

$$ \begin{cases} \dot{x}= \mu x-y + x(x^2+y^2)\\ \dot{y}=x+\mu y +y(x^2+y^2). \end{cases} $$

This time, a repelling limit cycle surrounding a source present for negative $\mu$ values collapses at the critical value $\mu=0$ to leave a sink for $\mu$ positive values.

The two previous examples were cooked up from our toy model of limit cycle. Fortunately, there is a general theorem telling us how to detect a Hopf bifurcation. We give here a soft version of it. Before stating the theorem, let’s precise the setting. Consider a system

$$ \begin{cases} \dot{x}=f(x,y,\mu)\\ \dot{y}=g(x,y,\mu) \end{cases} $$

where $\mu$ is a real parameter. For simplicity, assume that the functions $f$ and $g$ are smooth (all derivatives of $f$ and $g$ in $x,y$ and $\mu$ exist). Assume that there exists $\mu^*$ (the bifurcation parameter) such that $(\bar{x}(\mu^*),\bar{y}(\mu^*))$ is a fixed point for the system when $\mu=\mu^*$. By the implicit function theorem, there is a smooth curve of fixed points $(\bar{x}(\mu),\bar{y}(\mu))$ in an interval
containing the value $\mu=\mu^*$.

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.

Several remarks are in order:

  • The theorem tells something only local around the fixed point. For instance, in the supercritical case, maybe the limit cycle disapears when $\mu$ is above $\mu_{1}$, maybe not.
  • There is in fact a third possible scenario: infinitely many concentric cyles surrounding the fixed point can be born at the value $\mu=\mu^*$. But this situation is not generic.
  • There is a more powerful version of the theorem whose conclusion is that the system really behaves like our above two cooked-up examples, in the sense that it is
    topologically equivalent to one of them (up to the sense in which solutions rotate).
  • We’ve presented Andronov-Hopf bifurcation theorem for two-dimensional systems. In fact, there is a suitable version applicable to higher dimensional systems. This makes this theorem very attractive (compare with PoincarĂ©-Bendixson theorem that works only in dimension two).

An example of bifurcations of cycles

All the previous examples of bifurcations are local in the sense that they happen in the small neighborhood of a fixed point. Here we show examples of global bifurcations involving limit cycles.

Consider the two-dimensional system (given in polar coordinates)

$$ \begin{cases} \dot{r}=\mu r + r^3-r^5\\ \dot{\theta}=\omega + b r^2 \end{cases} $$

where $\mu<0$ will be the varying parameter. As we are going to see, when $\mu$ crosses $-1/4$, there is what is called a saddle-node bifurcation of cycles, to which $(0,0)$ does not participate (it remains a sink throughout). Watching what happens by decreasing $\mu$, we see a repelling and an attracting limit cycles that collide and disappear.