Abstract: We investigate the predictive power of recurrent neural networks for oscillatory systems not only on the attractor but in its vicinity as well. For this, we consider systems perturbed by an external force. This allows us to not merely predict the time evolution of the system but also study its dynamical properties, such as bifurcations, dynamical response curves, characteristic exponents, etc. It is shown that they can be effectively estimated even in some regions of the state space where no input data were given. We consider several different oscillatory examples, including self-sustained, excitatory, time-delay, and chaotic systems. Furthermore, with a statistical analysis, we assess the amount of training data required for effective inference for two common recurrent neural network cells, the long short-term memory and the gated recurrent unit.
Pyragas control is a time delayed feedback control scheme designed to stabilize periodic orbits. From an implementation perspective, its main advantage is that the only knowledge required is the period of the target periodic orbit.
For years, it was believed that periodic solutions with an odd number of Floquet multipliers cannot be stabilized with a Pyragas control scheme (the ‘odd number limitation’). Although true for the non-autonomous case [Nakajima, 1997], the claim has later been refuted for autonomous systems [Fiedler et al, 2008].
In this talk, we will have a look at the obstructions that result in the odd-number limitation in the non-autonomous case, and the mechanisms that refute it in the autonomous case. Moreover, we will discuss some generalisations that can overcome these obstructions.
Vafa-Witten invariants, introduced by Vafa and Witten in 1994,have recently been defined in algebraic geometry by Tanaka and Thomas, asthe virtual enumeration of certain sheaves on a local surface.We study contributions of sheaves in the counting problem, that can bedescribed by flags of torsion free sheaves of rank 1 on the underlyingsurface (the vertical contributions). We discuss auniversality result, which allows us to explicitly compute some of thesecontributions. The results can be used to verify precise formulas, conjecturedby Göttsche and Kool, for the vertical contributions to rank 2 and 3Vafa-Witten invariants.
A linkage in the plane is an n-gon with fixed sidelengths but free angles.
Question: What is the space of configurations of a given linkage?
Exercises: Answer the question for linkages whose sidelengths are
1 - (1,1,1)
2 - (3,1,1)
3 - (10,10,10,1)
4 - (20,10,10,1)
There is a surprisingly simple classification of these spaces which is based on a combinatorial formula for the Betti numbers. The technical tool is Morse theory.
Follow-up question: how many diffeomorphism types of manifolds are realizable as such a moduli space?
This question is harder than it looks if a precise answer is needed. However, we can find asymptotic bounds that show superexponential behaviour.