Time-delayed snapshots enable scientists to identify dynamics in chaotic systems

Many of the world’s most important systems, such as the atmosphere, turbulent fluids, and even the motion of the planets, behave unpredictably due to chaos and noise. Scientists often study these systems through their “random” measures, long-term statistical behaviors, rather than individual pathways. Although useful, these measures have a fundamental limitation: completely different systems may share the same statistics, making it impossible to identify the underlying dynamics.

Researchers led by mathematician Yunnan Yang have introduced a new way forward using time-delay snapshots. Their work, “Intrusive measures in time-delay coordinates for identification of unique dynamical systems,” was published in Physical review letters On October 17.

An invasive measure is a method of assigning a size or probability to parts of a system that does not change as the system changes or evolves. Time-delay snapshots use invariant measures expressed in time-delay coordinates—relating current observations to their past values ​​and providing enough information to distinguish between systems.

By translating these theoretical results into computational tools, the researchers were able to demonstrate their effectiveness with physical examples.

“These advances offer a robust method for uncovering the rules underlying complex phenomena, opening up new possibilities for weather forecasting, spacecraft design and the analysis of chaotic data in science and engineering,” said Yang, the Goenka Family Assistant Professor in Mathematics in the College of Arts and Sciences.

Jonah Botvinc Greenhouse, a doctoral candidate in the field of applied mathematics, was a co-author on the paper, as was Robert Martin, U-Dome Army Research Laboratory.

Yang said she was drawn to the subject because it was like solving a puzzle. “You’re given data that represents some basic physics or engineering quantity and it’s your job to really uncover the data and see what’s causing it. But the problem is when you can’t identify the quantity individually—there are two different models that give you the same data, so you can’t tell them apart. We need to have a unique representation, so that’s the motivation for our work.”

Their techniques can be applied to answer questions in biology, as living things change over time. In psychology, since humans change their behavior over time. In engineering, such as the drag of air flow on airplanes or traffic flow, and in other fields.

“We use time dynamic equations to model the underlying causes, and this can be as important as the probability of transmission of a virus like Covid,” Yang said.

The work took more than a year, but Yang says she was never afraid to get stuck. “Mathematicians are always tackling problems without answers. We love challenges.”