This article has been reviewed according to Science X's editorial process and policies. Editors have highlighted the following attributes while ensuring the content's credibility: Many systems in nature can settle into several different stable states, with the final state depending on their starting conditions. However, the boundaries separating these states are often far more complicated than they first appear.
Through new research published in Physical Review Letters, researchers in Ireland and Germany, led by Serhiy Yanchuk at University College Cork, have shown that these boundaries can contain narrow, hidden pathways, allowing systems to reach stable states from starting points that simpler models would rule out entirely. From signals in the brain to patterns in Earth's climate, many complex systems in nature can exist in more than one stable state—a property known as "multistability." The state such a system eventually settles into can vary depending on its underlying properties, but also heavily depends on where it started. This raises important questions about resilience: If a disturbance knocks a system away from its current state, will it return or tip into a different one entirely?
This kind of behavior is often seen in simplified models of complex systems, where a sudden switch between stable states is termed a "tipping point." On top of this, many complex systems also evolve on very different timescales at once. In Earth's climate, for example, fast-changing weather is coupled to far slower changes in oceans and ice sheets. To capture these effects in their models, physicists often use the idea of a "basin of attraction": the full set of starting points that eventually lead to a given stable state.
Drawing on this idea, Yanchuk's team developed new models of multistable systems containing both fast and slow processes. They then mapped out which starting conditions led to each stable state and simulated how the systems evolved over time. Stretching out from some basins, the researchers found long, narrow pathways that they called "singular funnels." These structures allow a system to return to a stable state from starting points far beyond where it would normally be expected to.
The team also showed that funnels become dramatically narrower as the gap between the fast and slow timescales widens—but as long as both processes run at finite speeds, they never fully disappear. As a result, a disturbance could switch a system between states in a way that can't be captured in simplified models. Yanchuk's team found these funnels in several different systems, from the simplest possible model with two competing states to complex networks of up to 10 linked oscillators.
This suggested that the structures could be a universal feature of complex systems. Ultimately, the findings offer a warning against drawing conclusions about resilience or "tipping points" from simplified models alone, including those used to assess the possible impacts of climate change. By accounting for these hidden funnels, researchers could now build more reliable models of the multistable systems around us—and, in turn, better predict how these systems will change.
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