Sunday, September 20, 2009

Atractors for the Differential Operator

The other day, I was thinking about how you can find fixed points for some functions in an easy way, for example, for $f(t)=\cos(t)$, if you start with any initial point $t_0$ and iterate the function on it, you will reach a fixed point, i.e. $f^n(t_0)\to t_{fix}$ as $n\to \infty$ which is approximately $0.739085$.

Then I ask myself if you can mimic the same procedure for finding fixed points, and hence attractors when instead of treating numbers one uses functions. In this particular case, I picked the differential operator $T=\frac{d}{dt}$, so the idea was to study the behavior of $\{T^n(x_0)\}_{n=0}^\infty$ for $x_0$ a initial function.

For instance, looking for the fixed points of this operator is a rather easy task, since it is the same as solving the differential equation $\frac{d}{dt}f(t)=f(t)$, which has general solution given by $f(t)=Ae^t$.

This approach give us more than just the fixed points for this operator, but also tells us that the space of functions in which we are interested for this operator to act, is the space of smooth funtions, i.e functions which have infinite number of derivates. Then we can formalize our discussion as studying the dinamics of the differential operator $T:C^\infty\to C^\infty$.

The natural question would be what happens to a general function when iterate $T$ to it? Is it going to be atracter by one of the fixed points? Or is it going to be blown up to "infinity"?

For example, if our initial function $x_0$ is a polynomial with degree $n$, we have that $T^{n+1}x_0=0$, so all polynomials are in the basin of atraction of 0, that is, after aplying many finite times the operator $T$, every polynomials will get map to 0.

On the other hand, if we start with $x_0=\cos(t)$ or $x_0=\sin(t)$, we have that this leads to a periodic orbit, namely $T^2x_0=x_0$, so the sequence $\{T^nx_0\}$ does not converge, but rather stays in a cycle.

In general, for having a periodic orbit when starting at a function $x$, one has to have that $T^mx=x$ for some $m\in\mathbb{N}$. Solving the differential equation gives that $x=\sum_{k=0}^{m-1}A_k e^{\lambda_m^k t}$, where $\lambda_m$ is the primitive $m$-root of unity in the complex numbers, and $A_k$ are constants.

Here we looked at some examples where the sequence converges or stays in a cycle, but it can also happen that the sequence diverges, for instance if we let $x_0=e^{cx}$, with $c\neq 1$, we will have that $\{T^nx_0\}$ diverges.

Now, if we try to analyze what the behaviour would be for a general smooth $x_0$ function, in order to stablish wheter the orbit of $x_0$ is attracted to a fixed point, it is periodic or it diverges, we need te notion of a distance in between functions or a norm. $C^\infty$ is a vector space, since the linear combination of smooth functions is again a smooth function, but it lacks the structure of a Banach space, which is that there doesn't exist any norm such that the space is complete. This doesn't help too much in pursue of a simple way of finding these attractors like we did with real and complex numbers via a analityc function. So the idea of finding these orbits for $T$ requires a little bit of more effort.

Trying to fix this impediment, I found that $C^\infty$ has the structure of a Fréchet space, with is endowed not with a norm, but instead with a family of seminorms. This way we can talk about being close to a fixed point and the notion of being attracted to.

I'm still reading at this point what can be done to generate this sets that are being attracted or repeled by this fixed points, which would be somehow equivalent to the notion of a Julia and Fatou sets for $T$. The idea of finding these fractals of functions seems to me really interesting and I haden't tought of it until I was actually writing this post, so it was somehow related to the previous post just by chance and it wasn't made on purpouse.

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