02/07/2020

Marcelo Viana discusses dynamical systems in a live broadcast by SBM.

Divulgação IMPA/ Tomás Rangel

IMPA's Director-General, Marcelo Viana, was the guest on this Wednesday's (1) live stream of the series “Getting to know the research areas in mathematics”, promoted by the Brazilian Mathematical Society (SBM). In the conversation, broadcast on the SBM's YouTube channel , Viana presented an overview of dynamical systems, an area of mathematics that he researches.

“We dynamicists tend to adopt a very ambitious view of the object of study of dynamical systems. A quick definition is that this is the area of mathematics that studies any phenomenon that evolves over time,” he said. “But everything around us is movement and evolution over time.”

The mathematician spoke about the origins of the field, considered young when compared to geometry, arithmetic, and algebra. "We would have to mention Newton, the creation of calculus, and also the development of the laws of mechanics." These laws showed that phenomena of physics and nature can be expressed through mathematical equations, more specifically through differential equations, he concluded.

But when we talk about dynamical systems as an autonomous area, developed with its own tools and ideas, its emergence is much more recent and is linked to the work of the French mathematician Henri Poincaré (1854-1912) on the theory of celestial mechanics, an area that studies the movement of planets around the sun. "He observed that the vast majority of differential equations could not be solved in the traditional sense of finding formulas and solutions. To describe the behavior of the solutions, it would be necessary to use a set of tools such as topology, geometry and algebra, and probability theory," explained Viana.

From this movement originated the qualitative theory of differential equations, the starting point for dynamical systems. "This helped to understand what the object of dynamical systems is. The area seeks to study phenomena that evolve over time, trying to understand this evolution, predict what will happen to the phenomenon in the long term and, at a certain point, control this evolution."

If one of the first problems of dynamical systems was the movement of planets around the sun, in Poincaré's time, after 100 years of advancement the area is currently applied in weather forecasting, modeling ecological systems, and communication networks, such as the internet. Its most current and contemporary use is in epidemiological modeling, Viana pointed out. “The models used by mathematicians to predict the evolution of the Covid-19 pandemic are based on ideas of dynamical systems. The situation and the number of infected people, the fatalities, all of these are dynamical variables that influence each other.”

Watch the full live stream: