A paper written by my flatmate has been accepted and he will present it in China, next October. To celebrate this wonderful event he decided to have a long weekend in Berlin.
I was busy working on another project and ordering some food from the internet (had to take advantage of the happy moment and ask some money to buy me drinks and junk food (oo)
But after two days I decided to meet him in Berlin, also because I finished all the food I bought in less than 24 hours…Hey! I was a hungry Pig, and I was super happy for him. This feeling normally makes me eat more.
I was so tired of going around Berlin, from one sightseeing bus to the other that I had to sit on a bench near Alexander Platz. I liked it there also because the people in the picture gave me food when I insisted that I was a bird too.
While they were concentrated into giving seeds and pieces of bread I realised something amazing. Those birds moved following trajectories I’ve been already seeing somewhere. It clearly was a strange attractor. They were attracted by seeds, but also scared by humans (and how they couldn’t! Sometimes I am scared t(oo)
It was very difficult to study the trajectories of those birds, also because I had no paper and no pen for some notes. By the way I remembered about a strange attractor which generates chaotic trajectories in a very similar way.
Otto Rössler came to my mind! Rossler, who – btw – was born in Berlin, is known for his contribution in chaos theory.
Attractors describe how a dynamical system “moves” in the plane (in our case it’s the xyz plane). Attractors are normally considered to explain very complex systems. Chaos is related to attractors for the reason you might understand.
He studied an attractor which follows an outward spiral on the (x,y) plane where an unstable fixed point is and when the system is “tired” to stay there, it follows another fixed point and rises to the z dimension.
Those birds were doing exactly the same: a fixed point on the (x,y) plane and another fixed point (the hands of the guy next to me and that tasty sandwich in the hands of the girl on the other bench) on the z dimension.
The Rossler’s attractor is defined by this set of non-linear differential equations
I simulated it on my computer with the parameters that Rossler studied at that time. So I set a=0.2, b=0.2, c=5.7 and started the computations.
Guess what I had? The birds’ trajectories (almost) (oo)
For those who feel curious like me Rossler’s attractor is amazing because it is formed by two out of three linear equations.
This looks weird. But it seems much more interesting if you think that a chaotic trajectory comes from such a simple system of equations. Moreover, those two linear equations can be studied in the (x,y) plane by setting the z coordinate to zero.
The resulting system will be:
For those who know a bit of dynamical systems, this is super easy. For those who are not into this topic just follow me (or quit reading with a “ooooh”)
The Jacobian matrix is a nice way to study the stability of a system. It basically takes into account the partial derivatives
which give an idea of how the system moves on the x and y coordinate. Theory says that if we have complex eigenvalues with positive real part, the equilibrium point will be an unstable spiral. There we go.
The eigenvalues of the system are 
If parameter a is in the range (0,2), these eigenvalues are complex and the origin is indeed an unstable spiral.
In the (x,y) plane (where z=0) this will lead to a spiral which gets bigger and bigger. But in the original system, z gets bigger and that will affect x too, which will decrease because of the negative contribute (look at the first equation). This makes the system going up and down in a weird way. Like the birds.
PS. (which clearly stands for Pig Scriptum)
Ok the birds here were just attracted by food. But it was cool to assume that they knew about the Rossler’s attractor
(oo)
