When you think of climate change, math might not be the first thing that comes to mind. But what if advanced mathematical tools could help scientists predict when one of Earth’s critical systems might hit a dangerous tipping point—one that could lead to abrupt and irreversible changes?* *That is exactly what Prof. Michael Ghil—a Distinguished Research Professor at UCLA and Distinguished Professor Emeritus at École Normale Supérieure in Paris—explored in his recent talk, *“Nonautonomous Dynamical Systems Help Study Long-term Trends and Abrupt Shifts in Climate Variability.”*

In this blog post, we will break down the key ideas from Prof. Ghil’s talk and show how math can be a powerful tool in our efforts to understand and predict climate tipping points. Prof. Ghil also advocates for the inclusion of mathematical techniques like algebraic topology in graduate school curricula, noting their significance in scientific breakthroughs, including award-winning research such as the Nobel prize in Physics in 2016 and the Abel prize in Mathematics in 2022.

**What Are Climate Tipping Points?**

Climate tipping points are critical thresholds in Earth’s climate system where small changes can trigger large, potentially irreversible shifts. For example, the Atlantic Meridional Overturning Circulation (AMOC), a crucial system of ocean currents, could reach a tipping point and collapse, leading to drastic changes in weather patterns and ecosystems. Recognizing and addressing these tipping points is essential to protect our future and build a more sustainable world.

**How Do Nonautonomous Dynamical Systems (NDSs) Work?**

So, what are these *nonautonomous dynamical systems* that Prof. Ghil talks about? Put simply, they are mathematical models that help scientists study complex systems—like Earth’s climate—that are constantly changing over time. In these models, "time-dependent forces" are always acting on the system. For example, greenhouse gas emissions from human activity are a time-dependent force that gradually increases and affects the climate.

Think of NDSs like the instructions in a GPS system that are continuously updated based on traffic conditions. These models allow us to account for natural and human-caused changes in the climate, and they help scientists understand how the climate will evolve under different scenarios.

**Why Are Attractors Important?**

Prof. Ghil explains that traditional climate models use something called an “attractor,” which can be imagined as a stable state that a system tends to fall into. But in real life, our climate does not behave so predictably because of time-dependent forces like human-produced concentrations of greenhouse gases, aerosols, or natural events such as volcanic eruptions. That’s why scientists now use *pullback attractors (PBAs)* and* random attractors (RAs)*—which take into account these ever-changing forces—to better predict the future behavior of the climate.

Here is a visual example of PBAs for a linear scalar ordinary differential equation (ODE), courtesy of Prof. Honghu Liu, Virginia Tech:

These attractors act like "magnets" that draw the climate system toward certain outcomes. By studying them, scientists can spot when the system is getting close to a tipping point—giving us a potential early warning before things take a drastic turn.

A fascinating example of a random attractor from the stochastic version of the Lorenz “butterfly” model shows that fractal structures can survive and even be influenced by noise.

**What Are Topological Tipping Points (TTPs)?**

One of the most exciting ideas in the talk is the concept of *Topological Tipping Points (TTPs)*. If you think of the climate system as a landscape with hills and valleys, a TTP is when the whole shape of the landscape changes, causing the climate to shift from one state to another. For example, a warming ocean could push ocean currents into a new, unstable pattern that drastically changes weather patterns around the world.

By using advanced mathematical tools from *algebraic topology*—a field of math that studies shapes and spaces—Prof. Ghil and his colleagues can map out these tipping points and identify when the climate is nearing a critical transition.

**Real-World Applications: Climate Variability and Ocean Currents**

To make these ideas more concrete, Prof. Ghil applies his models to real-world climate systems. He looks at the wind-driven double-gyre ocean circulation—a system of ocean currents in the North Atlantic and North Pacific Oceans. These currents help regulate global temperatures and weather patterns, so understanding how they might change under different conditions is crucial.

Prof. Ghil’s work also includes models of the mid-latitude atmosphere and seasonal weather patterns, showing how small changes can build up over time and lead to larger climate shifts.

**Why Does This Matter?**

You might be wondering why all this math matters. The answer is simple: understanding tipping points helps us prepare for climate change. If scientists can predict when Earth’s climate is nearing one of these dangerous turning points, governments and communities can take action—whether it is reducing emissions, preparing for more extreme weather, or planning for rising sea levels.

Prof. Ghil’s work is an important step toward giving us the tools to predict and hopefully prevent the worst outcomes of climate change. By using these advanced mathematical models, scientists can better understand the complex web of factors that influence our climate, and they can identify the moments when small changes could lead to large, irreversible impacts.

**Watch the Full Webinar**

Curious to learn more? Watch **Prof. Michael Ghil’s full talk** below, where he dives deeper into these mathematical concepts and shows how they are helping scientists predict long-term climate trends and abrupt shifts.

Also, you can download the **presentation slides** from the webinar** **to explore the models and concepts in more detail.

### Join Next Webinar

Tune in online for future ClimTip webinars on the first Tuesday of every month using this Zoom link. No registration needed. Visit our Youtube channel to access the webinars library.