Dr. Oded Zilberberg, who leads the Dynamics and Control of Quantum Materials group at the University of Konstanz, is at the center of new research showing that the behavior of nonlinear physical systems can be understood using ideas borrowed from something as familiar as a mountain landscape. His team, working with collaborators at ETH Zürich and CNR INO Trento, has introduced a framework that links the topology of a system’s dynamics to the layout of valleys, ridges and flow paths on a terrain.
Villa, G., del Pino, J., Dumont, V., Rastelli, G., Michałek, M., Eichler, A., & Zilberberg, O. (2025). Topological classification of driven-dissipative nonlinear systems. Science Advances, 11(33). https://doi.org/10.1126/sciadv.adt9311
The analogy begins with a simple picture. Standing on the peak of a mountain, you can trace how water would travel across the slopes, settle in valleys or diverge at ridges. These paths flow lines reflect the structure of the landscape. In physics, many driven-dissipative systems behave in a comparable way. The states in which such systems settle can be viewed as valleys, while unstable configurations resemble ridges. The paths that link these states act like the water’s route downhill. This overall pattern forms a kind of topological map describing how the system evolves without needing all the microscopic details.
Dr. Oded Zilberberg, from the University of Konstanz stated,
“For us, it is not just about identifying invariants, but about understanding how one stable configuration transforms into another.”
What makes the new work notable is that it extends ideas from topological physics typically used for linear systems into the nonlinear regime. Nonlinear oscillators, especially in micro-electromechanical systems, often support several stable oscillation modes. These modes can differ not only in amplitude but also in the direction in which their motion winds through phase space, a property known as chirality. Traditional tools do not easily classify such complex behavior, particularly when external driving or dissipation reshapes the system.
To address this gap, the researchers designed a topography-inspired method that builds a graph of the system’s stable points, unstable points and the boundaries between them. This graph acts as a topological invariant, meaning it stays consistent as long as the system’s structure does not undergo a fundamental change. When external parameters shift enough to reorganize the system’s dynamics similar to how an earthquake reshapes a mountain ridge the graph changes as well. This change marks a topological phase transition.
The team tested the framework using a nonlinear micro-electromechanical resonator. Through measurements of its ringdown behavior and detailed phase-space reconstructions, they identified multiple stable steady states and tracked how these states emerged, merged or vanished as the driving conditions were varied. The approach accurately captured transitions such as shifts between underdamped and overdamped behavior, as well as symmetry-breaking events associated with population inversion.
What strengthens the study is its combination of experimental work with structural analysis and theoretical modeling. High-resolution diffraction measurements confirmed how the underlying system geometry shapes the set of possible dynamical states, while a minimal mathematical model helped explain why certain transitions correspond to changes in topological structure rather than simple parameter drift.
The practical implications extend beyond academic classification. Many technologies rely on nonlinear systems, including sensors, photonic components, mechanical filters and emerging quantum devices. These systems can be sensitive to small changes in their environment. A topological framework that identifies which features of their behavior remain stable and which signals a change of regime could help engineers design components that are more reliable and easier to tune.
The researchers note that a major next step is adapting this approach to quantum systems, where fluctuations can obscure traditional signs of phase transitions. Early follow-up work suggests that key aspects of the flow-topology picture still hold in quantum settings, offering potential new tools for diagnosing transitions even when conventional theoretical markers vanish.
By treating dynamical behavior as a kind of landscape, this research provides a unified way to reason about systems that were previously difficult to categorize. It offers a path toward clearer understanding of how complex, nonlinear and dissipative systems change state — and may influence how engineers design the next generation of resilient, adaptable devices.
Adrian graduated with a Masters Degree (1st Class Honours) in Chemical Engineering from Chester University along with Harris. His master’s research aimed to develop a standardadised clean water oxygenation transfer procedure to test bubble diffusers that are currently used in the wastewater industry commercial market. He has also undergone placments in both US and China primarely focused within the R&D department and is an associate member of the Institute of Chemical Engineers (IChemE).
