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 PoincarĂ©-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. HenceWe 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
*

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

*with*

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

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.