Time-delay 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 planetary motion, behave unpredictably due to chaos and noise. Scientists often study these systems using their “invariant” measures, long-term statistical behavior, rather than individual pathways. Despite their usefulness, these measures have a fundamental limitation: completely different systems may use the same statistics, making it impossible to determine the underlying dynamics.

Researchers led by mathematician Yunan Yang have proposed a new way forward using time-lapse images. Their work “Invariant measures in time-delayed coordinates for unique identification of dynamical systems” was published V Physical Review Letters October 17.

An invariant measure is a way of assigning a size or probability to parts of a system that remain the same as the system transforms or evolves. Lag snapshots use invariant measures, expressed in lag coordinates, that relate current observations to their past values ​​and provide enough information to distinguish between systems.

Translation of these theoretical results into computing toolsThe researchers were able to demonstrate their effectiveness using physical examples.

“This breakthrough offers a robust method for uncovering the rules underlying complex phenomena, opening new possibilities for problems such as weather forecasting, spacecraft design and analysis of chaotic data in science and engineering,” said Yang, Goenka Family Assistant Professor of Mathematics in the College of Arts and Sciences.

The paper was co-authored by Jonah Botvinick-Greenhaus, a doctoral student in applied mathematics, and Robert Martin of the Army Research Laboratory U DEVCOM.

Yang said she was drawn to the topic because it felt like solving a puzzle. “You're given data that represents some basic physical or engineering quantity, and your job is to really unravel that data and see what's causing it. But the problem is that you can't uniquely identify the quantities – there are two different models that give you the same data, so you can't tell them apart. We need to have a unique perspective and that is the motivation for our work.”

Their technique can be applied to answer questions in biology because living things change over time; in psychology, because people change their behavior over time; in technology, for example, with air flow resistance in airplanes or traffic, and in other areas.

“We use time history equations to model root causes, and this can be as important as the transmission probability of viruses like COVID,” Yang said.

The work took more than a year, but Young says she was never afraid of getting stuck. “Mathematicians always deal with problems that have no answer. We like a challenge.”