Dynamical systems are mathematical models that describe the evolution of systems with time, such as the motion of a pendulum, the changes of a fish population, the concentration of a certain chemical or the electric variations inside a neuron. At any given moment, the state of a dynamical system is identified with a point in a state space. By studying the geometry of this space, we can determine qualitative aspects of the evolution of the system. Some typical questions that we try to answer are:
- Are there any stationary states?
- Can we find some periodic behaviour?
- What does the system look like after a great amount of time has passed?
Completely integrable systems
In certain occasions, some physical quantities such as the energy are preserved. This simplifies the study of these systems, since we know that solutions must stay within the same energy level. This idea can be generalised to systems with 2n-dimensional state spaces and n conserved quantities, known as completely integrable systems. Harmonic oscillators, the spherical pendulum or the Euler top are all examples of this type.
In line with M. Kac’s famous paper entitled “Can we hear the shape of a drum?”, one can ask how much of the properties of an integrable system are retained in its energy values. A good example is that of so-called toric systems, for which the conserved quantities define an effective n-dimensional torus action. T. Delzant showed that these systems can be completely classified in terms of certain polytopes, which correspond to the set of possible energy values.
Inspired by this situation, Á. Pelayo and S. Vũ Ngọc found in the recent years a global classification for a certain class of integrable systems with a 4-dimensional state space, called semitoric systems. These systems have two compatible conserved quantities, but only one of them comes from a rotation symmetry. The classification is based on five invariants and it is relevant because such systems appear often in physics, among others as classical counterparts of quantum systems.
My current research consists of the computation of these invariants for some families of systems that depend on several parameters with the help of powerful mathematical software such as Mathematica. We have also shown how the invariants convey this dependence on the parameters and also how they display the symmetries of the systems. Together with my collaborators in Australia, Belgium and the US, we aim at a better understanding of both semitoric systems and the nature of their classification invariants.