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

Many of the planet’s most important systems, such as the atmosphere, turbulent fluids, and even the movement of planets, behave unpredictably due to chaos and noise. Scientists often study these systems through their “invariant” measurements, their long-term statistical behaviors, rather than their individual trajectories. While useful, these measurements have a fundamental limitation: Completely different systems can share the same statistics, making it impossible to identify the underlying dynamics.

Researchers led by mathematician Yunan Yang have introduced a new way forward, using delayed snapshots. Their work, “Invariant Measurements in Time Coordinates for Unique Dynamic System Identification,” was published in Physical Examination Letters on October 17.

An invariant measure is a way of assigning a size or probability to parts of a system that remain unchanged when the system transforms or evolves. Delay snapshots use invariant measurements expressed in delay coordinates (relating current observations to their past values) and providing sufficient information to distinguish systems.

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

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

Jonah Botvinick-Greenhouse, a doctoral student in the field of applied mathematics, was a co-author of the paper, as was Robert Martin of the U DEVCOM military research laboratory.

Yang said she was drawn to the subject because it’s like solving a puzzle. “You’re given data that represents some underlying physical or engineering quantities and it’s up to you to really unravel the 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 a unique representation, so that’s the motivation for our work.”

Their technique can be applied to answer biological questions, since living beings evolve over time; in psychology, since humans change their behavior over time; in engineering, such as airflow drag in airplanes or traffic flow, and in other fields.

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

The work took more than a year, but Yang says she was never afraid of getting stuck. “Mathematicians are always faced with unanswered problems. We like challenges.”