# Introduction

I have changed my name to Dan Lilja. My new website can be found at www.danlilja.se.

I'm a PhD student in mathematics at the Department of Mathematics of Uppsala University, where I'm also a member of the CAPA group. In addition to my PhD studies I am currently a board member of the Mathematical Society and the PhD student representative on the board of the Department of Mathematics.

My area of research is called *dynamical systems*. In particular I
study renormalization of exact symplectic twist maps of the plane and
their invariant Cantor sets. Outside of dynamical systems I also have an
interest in geometry, topology and mathematical physics.

## Dynamical systems

Dynamical systems describe how objects and their properties change with respect to time. For example it can describe the motion of a particle under the influence of a force such as gravity, how the concentrations of various substances change during a chemical reaction or how the number of fish in a lake changes from year to year.

Dynamical systems are usually divided into two categories depending on
how time is modeled in the system: *continuous* and
*discrete*. The first two examples above are examples of continuous
dynamical systems since time is considered as a continuum. There are no
holes. The last example is a discrete dynamical system since the
number of fish is only given at discrete times, namely once a year.
Mathematically a dynamical system consists of a space \(M\) called the
*phase space* and a function \(F\) taking as its arguments a time
\(t\) and a point \(x\) of phase space and outputs another point
\(y=F(t,x)\) of phase space, the point to which \(x\) has moved to after
time \(t\). We therefore also require that \(F(0,x)=x\) for every point
\(x\), i.e. if no time passes then the points do not move. Discrete
dynamical systems are usually just considered to be a function \(f\)
taking points of phase space to other points of phase space and time is
considered as integers giving the number of iterations of this function.
If the point \(x\) is mapped to the point \(y\) after time \(t=n\) we
write \(y=f^{n}(x)\) where \(f^{n}\) denotes \(n\) iterations of \(f\).

## Chaos

Many nonlinear dynamical systems, for example most dynamical systems
coming from applications, display a phenomenon called *chaos*.
Examples of dynamical systems exhibiting chaotic behaviour include the
Lorenz system describing atmospheric convection, the Belousov-Zhabotinsky
chemical reaction, Chua's circuit, the kicked rotator and the double
pendulum.

One common mathematical definition of a chaotic dynamical system states that a dynamical system is chaotic if it has three properties called sensitive dependence on initial conditions, topological mixing and if periodic points are dense.

*Sensitive dependence on initial conditions* means that no matter
how close you look at any particular point you will always find other
points that eventually move a fixed distance away from the point you're
looking at. In this way it is a mathematically precise definition of the
famous butterfly effect. As an illustration consider the circle in the
plane. We can describe any point on this circle by giving the angle
between the point and the \(x-\)axis counted counterclockwise. We can
then consider the discrete dynamical system given by doubling the angle,
called the *doubling map*. Then no matter how close two points are
to each other the angle between them will double with each iteration. So
for any fixed angle, say 90 degrees, the angle between the two points
will be larger than 90 degrees.

*Topological mixing*, also called *topological transitivity*,
means that if we pick two nonempty, open sets and apply the dynamical
system to one of them then it will eventually intersect the other set.
This condition is a formal way of saying that any collection of points
will eventually be spread throughout phase space. Our example of the
doubling map of the circle is topologically mixing. If we start
with any arc along the circle it will double in size with each iteration
of the doubling map. No matter how small the arc we choose is it will
eventually cover the entire circle and will therefore intersect any other
arc as well.

Lastly, *periodic points being dense* requires some explanation.
First, a point \(x\) of phase space is called *periodic* if there is
some time \(T\) such that \(F(T,x)=x\). This means that the point will
repeatedly move around for a while in a specific orbit and then come back
to its starting point. That such points are *dense* means that
whichever point of phase space we pick we can always find a periodic point
arbitrarily close to it. No matter how much we zoom in on our chosen point
we will always see periodic points as well. Compare this with how the
rational numbers are distributed within the real numbers. The periodic
points of the doubling map are dense in the circle. To see this requires
first that rational numbers are dense in the real numbers and that a
rational number has an eventually periodic binary expansion in the same
way as their decimal expansion is always eventually periodic. The
doubling map corresponds to shifting the binary expansion by one place
and so that every point on the circle with a rational angle is eventually
periodic. In conclusion we have informally proven that the doubling map
is a chaotic dynamical system, giving us yet another example.

## Further reading

Here are some links for those that are interested in reading more about dynamical systems, chaos, renormalization, or my field of research.