# An afternoon on maths and dynamics: 3 public talks on delays, earthquakes and mammals

The event is organized by the Scuola Superiore of the University of Udine on the occasion of the IFAC TDS 2024 workshop.

### Program and abstracts

- 16:00 Introduction
- 16:05–16:50
*Sabine Mondié (CINVESTAV, Mexico City, Mexico)*

**Delays in biology and technology: Why do researchers study this topic?**

The aim of this talk is to introduce the topic of time delay systems to a non-expert audience. In the first part, the presence and impact of delays in systems are explained via several illustrative examples from biology and technology. In the second part, a discussion is given on how researchers develop tools to overcome the problems raised by delays or deliberately introduce delays to improve systems behavior. - 16:55–17:40
*Hinke Osinga (University of Auckland, New Zealand)*

**Maths on shaky ground**

Earthquakes can cause substantial damage to buildings in ways that are still not well understood. The amplitude and principal frequency of an earthquake are two primary components that affect the extent of the damage, and they are the basis for many design specification guidelines. In other words, the earthquake is modelled as a pure sine wave! As a case study, we consider a model of a planar, post-tensioned frame that exhibits dynamics quite similar to the experimental measurements of a scaled model on a shake table; the post-tensioned frame is a common feature in low-damage seismic design. The frame remains structurally sound as long as the tilt angle of the frame does not exceed a certain maximum. Many results in the literature are obtained from performing a large number of simulations over a range of wave amplitudes and frequencies. Such a brute-force approach establishes a region in the frequency–amplitude plane for which the structural stability of the frame eventually fails. Our approach is much more efficient, because we apply a novel computational method that determines the failure boundary directly. This method is based on continuation of a suitable two-point boundary value problem. Our computations demonstrate that, despite the idealised sinusoidal earthquake, the failure boundary is only piecewise smooth, and the results highlight further interesting details of how the dynamics is organised in the frequency–amplitude plane. We also address the elephant in the room: “Is the purely sinusoidal harmonic wave a good approximation of a real earthquake?”. We investigate how our results compare when the earthquake is modelled as an aperiodic wave with a broader frequency content. - 17:45–18:30
*Stefano Marmi (Scuola Normale Superiore, Pisa, Italy)*

**Dynamique of small mammal populations, seasonality and strange attractors**

One of the most striking phenomena, both for its spectacular nature and its importance in the history of ecology, is that of the cyclical variations of certain populations of small mammals such as voles or lemmings. These small rodents, at least in certain northern regions, have the particularity of “proliferating” every 3 to 5 years, which allows many emblematic species of these regions, such as snowy owls or polar foxes, to reproduce. These fluctuations are statistically periodic, but their amplitude can vary greatly, and the cycles can sometimes practically disappear for a few decades. Even if it is not difficult to obtain cycles with, for example, predator–prey models (like in the classical Lotka–Volterra model), an important open problem was to know whether mechanisms intrinsic to the population could also generate cyclical variations. The positive solution was given by the Field medalist Jean-Christophe Yoccoz in his only (unpublished) applied math work, by constructing a dynamical system with delays and seasonal effects which has a strange attractor and generates the desired (approximately) cyclical variations.