Constructing a 3-dimensional differential equation with no equilbrium points, but a stable limit cycle.

Today I wanted to share a fun dynamical system. To give some context, in a two dimensional setting, there must be a equilibrium in the interior of said limit cycle. This is a consequence of the Poincaré–Bendixson theorem. Here I would like to stress that we are dealing with a continuous differential equation, where there are no piecewise functions allowed. However in the third dimension there is no Poincaré–Bendixson theorem. This got me wondering if there is such a system where there is  no equilibria at all, but there is a stable limit cycle. As we will see, the answer is yes!

 

To construct such a system, I find working with cylindrical coordinates is easiest. The reason is we can work in the $r$-$z$ plane, then rotate around the $z$-axis to create our limit cycle. This is because any equilibrium points in the $r$-$z$ plane that are not on $r=0$ will become a limit cycle as they rotate around the $z$ axis. However, any equilibrium point on $r=0$ will be a fixed point. Thus, we need to accomplish two things, make sure that any point on $r=0$ does not stop moving, and that any point near $r=0$ moves away from $r=0$. Accomplishing this is not too difficult.

 

$$\frac{dr}{dt} = r f(r,z)$$

$$ \frac{dz}{dt} =k+r g(r,z)$$

$$\frac{d\theta}{dt} = \Omega$$


For suitably well defined functions $f$ and $g$ this basic equation will meet the conditions provide that $k\neq0$, then any solution that starts on $r=0$ stays on $r=0$ and moves up or down at a constant rate. Moreover, any equilibrium point of the $r$-$z$system will become a limit cycle.

 The next part of the construction requires us to find a system with a single equilibrium point. I am going to construct a system, where the equilibrium point undergoes a supercritical Hopf bifurcation. I did this mostly because it creates a cool torus solution as a byproduct.

 Recall that the normal form Hopf Bifurcation looks likes  

$$\frac{dR}{dt} = a R - \omega z - R(R^2+z^2)$$

$$ \frac{dz}{dt} =\omega R + a z - z(R^2+z^2)$$

If we let $R= r-r^*$ then and let $f(r,z)  = a R - \omega z - R(R^2+z^2)$ and $g(r,z)= \omega R + a z - z(R^2+z^2)$, we get the following system.


$$\frac{dr}{dt} = r [a (r-r^*)- \omega z - (r-r^*)((r-r^*)^2+z^2)]$$

$$ \frac{dz}{dt} =k+r [\omega (r-r^*) + a z - z((r-r^*)^2+z^2)]$$

$$\frac{d\theta}{dt} = \Omega$$

Here, whenever $a<0$ we have one stable spiraling equilibrium point in the $r$-$z$, that spirals inward at the rate $\omega$. Furthermore, this $r$-$z$ equilibrium point is smeared out when rotated around the $z$-axis, forming  a limit cycle with radius $r^*$ that rotates at the rate $\Omega$.

An example of the stable limit cycle for $a<0$. Pink and Black are different initial conditions. Both spiral downward towards the limit cycle. 

However, I find the solution with $a>0$ to be more fun. Recall, I used the normal form for a supercritical hopf bifurcation. Therefore, with positive $a$ there exists a limit cycle in the  $r$-$z$plane. This limit cycle gets swept out as an attractive toriodal surface with quasi periodic orbits. Inside the tours will be an unstable limit cycle, who has solutions spiral outward from it towards the surface of the torus.  These orbits will only be limit cycles in the 3D system iff $\frac{\Omega}{\omega_c}$ is a rational number. Here $\omega_c$ is the frequency of the $r$-$z$plane's limit cycle. I denote it with the subscript $c$ because I am unsure if it is equal to $\omega$.

Top: The Stable Torus appears for $a>0$.  Bottom: The $r$-$z$ plain with a stable limit cycle that becomes the stable torus. Yellow and blue lines are the nullclines for $r$ and $z$ respectively.

 I hope you find this construction useful. I find little exercises like this great practice for dynamical systems. Understanding how to construct systems like is sort of like playing with Legos. Knowing what geometric pieces to put together is key. Practicing with fun problems like this can strength your ability to construct and tune dynamical systems. 

If you want to read more about how dynamical systems is used in neuroscience, I suggest you read a few other of our blog articles, on simple neural models, NMDA spikes, cardiac dynamics, and dynamical bananas!

 
Author: Alexander White