Earth's climate is a complex system with more than one stable state. For 11,700 years it has rested in one of them — the Holocene — the only state in which civilization has ever existed. This page tells the same story in three voices: Physics (the Hamiltonian view: basins, ridges and a ball you can push), Sequence (the Schefferian view: the six beats by which one regime gives way to another), and Trajectory (the Waddingtonian view: the planet's path through that landscape). Three scrollable companions carry on from here: Resilience grounds all three in the free-energy mathematics of Bertolami & Nyström (2026), Early warning shows what an approaching tipping point looks like in the data, and Transformation asks what happens once the ground has given way.
Show the introduction again on next visit Why the same picture keeps reappearing A ball rolling between valleys — that single image was reached, independently, by physicists studying mechanical stability, by Conrad Waddington watching cells choose their fates, by C.S. Holling counting spruce-budworm outbreaks, by John Hopfield modelling memory in neurons, and by Earth-system scientists watching ice sheets thin. When five disciplines converge on the same shape, the shape is telling us something real about how complex systems work.
Whenever a system has more than one way to be stable, the mathematics tends to look like an energy landscape with valleys (stable states), ridges (thresholds) and a moving point (the system itself). That this picture is the same one for cell differentiation, lake eutrophication, memory recall and climate tipping points is itself one of the deepest unifying insights of complex-systems science.
Origins of the stable-basin analogy The mechanical idea is old. Helmholtz , Maxwell , and Poincaré formalised the picture of a system as a marble in a potential well in the late 1800s; equilibrium was a minimum of energy, stability was the curvature of that minimum.
The first leap outside physics was made in 1940 (and developed in 1957) by the British biologist Conrad Waddington . To explain how a single fertilised egg could become many distinct cell types, he drew an "epigenetic landscape" of branching valleys: a marble rolling down would end up in one lineage or another depending on small biases along the way. The picture stuck.
In the 1970s, René Thom and Christopher Zeeman 's catastrophe theory gave the geometry of those landscapes a rigorous classification: folds, cusps, swallowtails.
Ecology arrived by way of three papers that Scheffer himself names as the pioneering ones. Richard Lewontin (1969) set out the theoretical possibility that an ecosystem could have more than one stable state. C.S. Holling (1973) tied that theory to practical ecology and gave it the name we still use — resilience : not how fast a system bounces back, but how big a disturbance it can absorb before it ends up somewhere else. Robert May (1977) brought thresholds and breakpoints to a wide audience, and Imanuel Noy-Meir (1975) showed the same geometry in overgrazed rangelands with nothing more than two curves on a graph.
From the 1990s onward, Marten Scheffer turned the cup-and-ball into the canonical diagram of regime shifts in lakes, savannas, coral reefs, and finally the Earth System itself (Scheffer 2001, 2009). Around the same time, John Hopfield (1982) showed that the same energy-landscape mathematics describes how neural networks store memories — each saved pattern is a basin the network rolls into. Today, Lenton , Steffen , and colleagues use the same picture to chart the trajectory of the whole planet.
Physics — the Hamiltonian view Imagine Earth's climate as a ball resting in a valley. For over 10,000 years, the Holocene has been that valley: a stable climate state where ice sheets, ocean currents, monsoons and ecosystems found a dynamic equilibrium. The ball sits at the bottom, gently jiggling from natural variability — volcanic eruptions, solar cycles, orbital shifts — but always rolling back to the centre. That is resilience: how large a push the system can absorb without leaving its stable state — the room between where it rests and the ridge, not the drop beneath it.
Since the mid-20th century, human activity has become a sustained perturbation on a scale the Holocene basin has never experienced. Greenhouse-gas emissions, deforestation and industrial agriculture are not gentle nudges — they are a constant uphill push. As the ball climbs the basin wall, it approaches the ridge that separates the Holocene from a second attractor: a Hothouse Earth state, several degrees warmer, with higher seas, rearranged biomes, and feedback loops that would sustain the new climate long after emissions stop.
The ridge between the two basins is the critical threshold. Its height is climate resilience — the amount of forcing the system can absorb before tipping. But resilience is not fixed: each tipping element that activates (permafrost thaw, Amazon dieback, ice-sheet collapse) lowers the ridge. At the same time, rising emissions push the ball harder.
Why "Hamiltonian"? The Hamiltonian formulation of mechanics was introduced by William Rowan Hamilton in 1833. Instead of tracking forces and accelerations, it describes a system through its total energy — expressed as a function H(q, p) of position q and momentum p — and lets the dynamics fall out as flows on this energy surface. The same machinery underlies celestial mechanics, optics, quantum theory and statistical physics. Climate scientists borrowed the language to describe systems that can settle into more than one stable state, and the stochastic version (the Langevin equation used here) adds a noise term for volcanism, internal variability and human forcing.
How to use this view Drag to rotate the landscape; scroll to zoom.Perturbation : strength of random forces jostling the system.Resilience : ridge height between the two basins. Lower it to simulate cascading tipping points.Distance : how far apart the two attractors sit in state space.Holocene / Hothouse sliders: independent depth of each attractor.Flow : density of the streamlines drawn on the floor. They are coloured by the attractor they belong to — teal for the Holocene basin, amber for the Hothouse. The mathematics H(q, p) = p2 /(2m) + V(q)
V(q) = a(q2 − q0 2 )2 − dL G(q, −q0 ) − dR G(q, +q0 )
dq/dt = ∂H/∂p · dp/dt = −∂H/∂q − γp + σξ(t)
q = climate state, p = momentum, γ = friction, σ = noise (perturbation), ξ(t) = random forcing, G = Gaussian well for independent basin depths.
Strictly, the third line is where this stops being Hamiltonian. A Hamiltonian system conserves H; the −γp term is what spends it, and the noise term is what tops it back up. What is drawn here is a damped, driven system moving on a Hamiltonian's energy surface — the name belongs to the landscape, not to the dynamics. The distinction is not pedantry: the usual model in the tipping-point literature is this equation with the momentum removed altogether, where the slope of the landscape gives the speed directly rather than the acceleration. That is what the early-warning page integrates, and what the Schefferian view next door does. Keeping inertia here is a deliberate choice — it is what lets you push the planet and watch it swing.
Sequence — the Schefferian view This view adapts what the Dutch ecologist Marten Scheffer calls "perhaps the icon of the theory of alternative stable states" — the stability landscape. Its specific construction, a row of landscapes standing above the folded equilibrium curve they belong to, is figure 2.6 of his 2009 book Critical Transitions in Nature and Society ; the paper that put the picture in front of everyone is Scheffer, Carpenter, Foley, Folke & Walker , "Catastrophic shifts in ecosystems", Nature 413 (2001).
Six beats walk through the lifecycle of a regime shift, projected over a biome that itself transforms from humid forest to savanna to desert — the case Scheffer opens his book with, the greening and then abrupt loss of the Sahara around 5,500 years ago.
One thing this view does not do: the ball has no momentum. A stability landscape is not a hill — its slope is the system's rate of change, not a force — so a ball with inertia would read the picture one derivative out and could sail through a ridge unpushed. Scheffer devotes a section to the point, asking you to imagine the ball "rolling through a very viscous fluid". It glides here, and stops when the ground goes flat. The Hamiltonian view next door keeps its momentum on purpose; that one is a mechanics analogy and says so.
What the six beats show Beat 1 — Regime 1. A single deep valley. The system rests in one stable state and bounces back from everyday noise.Beat 2 — Perturbation. Sustained pressure begins to reshape the landscape itself. A second hollow appears beside the first.Beat 3 — Regime 2 deepens. The new attractor grows. Bistability is the hallmark of a regime-shift system.Beat 4 — Resilience boundary. The two basins are equal. The dashed arc traces the climb from the resting state to the ridge — the size of perturbation needed to flip the system, which is what resilience measures.Beat 5 — Tipping point. Pushed past the ridge, the ball stalls at the saddle. The slightest forcing now decides where it falls.Beat 6 — Regime shift. The system has fallen into the second basin. The original valley fades and may not return. What "resilience" really means There are two answers, and the difference between them is the whole subject. The everyday one is how quickly a system recovers from a knock — what Holling called engineering resilience . The one that decides whether a system survives is how large a knock it can take before it ends up somewhere else and stays there — ecological resilience , and in this picture that is the width of the valley, the room between where the system rests and the ridge at its edge.
A forest can look healthy right up to the moment it isn't. Two ecosystems that appear identical can have very different resilience: one with room on both sides, one perched a step from the ridge. What we want is not a quick recovery, but room to spare.
The two are not rivals, though, and that turns out to matter enormously: in most models they move together. As the room around a system shrinks, its recovery slows. That correlation is the reason the next paragraph works at all — it is what lets us measure something we cannot see by watching something we can.
How we know we're near a tipping point This is where the two definitions above pay off. As a basin flattens, the ball takes longer to settle after each push — so the recovery rate we can measure tracks the room to spare that we cannot . This phenomenon — called critical slowing-down — produces measurable statistical signatures: rising variance, increasing autocorrelation, a memory that lingers too long. The same fingerprint has been read in lakes about to flip, in coral reefs, in past climate transitions buried in ice cores — and, increasingly, in parts of the present Earth System. These early-warning signals are not predictions of when , but evidence of how close .
Trajectory — the Waddingtonian view Imagine the Earth System as a marble rolling across a hilly landscape. Valleys are stable climate states; ridges are thresholds it has to cross to switch from one state to another. The shape itself is set by the interplay of ice, oceans, atmosphere, biosphere — and now humanity.
For nearly three million years, the marble has rolled around in the cool Pleistocene side of the landscape — long glacial periods broken by warmer interludes. Around 11,700 years ago it settled into a particularly stable hollow, the Holocene, and stayed there. All of human civilization unfolded inside that one valley.
Today, sustained emissions and land-use change are rolling the marble up and out of the Holocene basin. We are at the rim — the present-day saddle. Just beyond it, the landscape forks. One path slopes gently sideways to a managed valley, Stabilized Earth , that humans hold open through deliberate stewardship. The other drops away into a deep, irreversible attractor, Hothouse Earth .
Why "Waddingtonian"? The metaphor was coined in 1957 by the British developmental biologist Conrad Hal Waddington in The Strategy of the Genes . To explain how a single fertilised cell could differentiate into many distinct cell types, he drew an "epigenetic landscape" of branching valleys. Half a century later, the same picture turned out to fit a much larger system: lakes flipping between clear and algae-dominated states, savannas tipping into desert, coral reefs collapsing into algae fields — and finally the whole Earth System. The 2018 paper by Steffen and colleagues, on which this view is based, brought the landscape metaphor explicitly into planetary-scale climate policy.
How to use this view Drag to rotate; scroll to zoom; hover any label for a plain-language explanation.The marble enters from the Pleistocene plateau, traces glacial cycles, slips into the Holocene basin, and climbs to Present day . Choose Stabilized Earth to watch a beam of stewardship divert the marble sideways. Choose Hothouse Earth to watch the marble cross the planetary tipping point and cascade. The physics of a safe harbour In 2026 the physicist Orfeu Bertolami and the resilience scientist Magnus Nyström gave this landscape an equation. Resilience, they show, has two physical ingredients. The first is a metastable state — a valley the Earth can be held in, part-way between the Holocene and the Hothouse, that has to be actively created and then maintained. The second is dynamic friction — a drag that resists the slide from one state to another; the same friction the Hamiltonian view already uses. Without both, sustained human pressure makes the runaway to a Hothouse state unavoidable. With them, the trajectory can be held bounded — which is what “stewardship” means in physical terms: opening a valley, and supplying the friction to stay in it.
In this view you watch that play out: human pressure — the combined weight of the nine planetary boundaries, seven of them already crossed — tilts the landscape and rolls the marble up out of the Holocene, and the stewardship step is the moment the metastable valley is opened beneath it.
Cousins from complexity science Regime shifts sit inside a wider family of ideas about how complex systems behave when pushed.
Ilya Prigogine (Nobel 1977) — dissipative structures : open systems far from equilibrium spontaneously organise into ordered patterns and can reorganise abruptly when conditions change.Robert May (1976) — showed that even simple ecological models can produce chaos and multiple stable states; bridge between mathematics and ecology.Stuart Kauffman — NK landscapes and "order for free": fitness landscapes with many peaks behave like the energy landscape used here, but for evolution.Per Bak — self-organised criticality : many natural systems (sandpiles, earthquakes, forest fires) tune themselves to the edge of a tipping cascade.Steven Strogatz — Nonlinear Dynamics and Chaos (1994): the textbook on bifurcations, the formal name for what happens when a basin appears, deepens, or vanishes.Duncan Watts & Albert-László Barabási — network resilience: how the topology of connections decides whether shocks fade or cascade.Dirk Helbing — systemic risk : how interconnected technical and social systems can flip into failure modes that no single component is responsible for. Scientific references Resilience & regime shifts Holling, C. S. (1973). Resilience and stability of ecological systems. Annual Review of Ecology and Systematics , 4, 1–23. May, R. M. (1977). Thresholds and breakpoints in ecosystems with a multiplicity of stable states. Nature , 269, 471–477. Scheffer, M., Carpenter, S., Foley, J. A., Folke, C., & Walker, B. (2001). Catastrophic shifts in ecosystems. Nature , 413(6856), 591–596. Carpenter, S. R. (2003). Regime Shifts in Lake Ecosystems: Pattern and Variation. Ecology Institute. Folke, C. (2006). Resilience: the emergence of a perspective for social–ecological systems analyses. Global Environmental Change , 16(3), 253–267. Walker, B. & Salt, D. (2006). Resilience Thinking: Sustaining Ecosystems and People in a Changing World. Island Press. Scheffer, M. (2009). Critical Transitions in Nature and Society. Princeton University Press. Dakos, V., Scheffer, M., van Nes, E. H., Brovkin, V., Petoukhov, V., & Held, H. (2008). Slowing down as an early warning signal for abrupt climate change. PNAS , 105(38), 14308–14312. Dakos, V., Carpenter, S. R., Brock, W. A., Ellison, A. M., Guttal, V., Ives, A. R., Kéfi, S., Livina, V., Seekell, D. A., van Nes, E. H., & Scheffer, M. (2012). Methods for detecting early warnings of critical transitions in time series illustrated using simulated ecological data. PLOS ONE , 7(7), e41010. Earth-system tipping Lenton, T. M., Held, H., Kriegler, E., Hall, J. W., Lucht, W., Rahmstorf, S., & Schellnhuber, H. J. (2008). Tipping elements in the Earth's climate system. PNAS , 105(6), 1786–1793. Steffen, W., Rockström, J., Richardson, K., Lenton, T. M., et al. (2018). Trajectories of the Earth System in the Anthropocene. PNAS , 115(33), 8252–8259. DOI Lenton, T. M., Rockström, J., Gaffney, O., Rahmstorf, S., Richardson, K., Steffen, W., & Schellnhuber, H. J. (2019). Climate tipping points — too risky to bet against. Nature , 575, 592–595. Armstrong McKay, D. I., Staal, A., Abrams, J. F., Winkelmann, R., et al. (2022). Exceeding 1.5°C global warming could trigger multiple climate tipping points. Science , 377(6611). Ritchie, P. D. L., Clarke, J. J., Cox, P. M., & Huntingford, C. (2021). Overshooting tipping point thresholds in a changing climate. Nature , 592, 517–523. Bertolami, O. & Nyström, M. (2026). Setting up the physical principles of resilience in a model of the Earth System. Earth System Dynamics , 17, 141–150. DOI Möller, T., Hoghöj, N. B., Wunderling, N., et al. (2024). Achieving net zero greenhouse gas emissions critical to limit climate tipping risks. Nature Communications , 15, 6192. DOI Ripple, W. J., Wolf, C., Rockström, J., Richardson, K., Wunderling, N., Gregg, J. W., Westerhold, T., & Schellnhuber, H. J. (2026). The risk of a hothouse Earth trajectory. One Earth . Lineage & complexity Waddington, C. H. (1957). The Strategy of the Genes. Allen & Unwin. Thom, R. (1972). Stabilité structurelle et morphogénèse. W.A. Benjamin. Zeeman, E. C. (1976). Catastrophe theory. Scientific American , 234(4), 65–83. Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective computational abilities. PNAS , 79(8), 2554–2558. Prigogine, I. & Stengers, I. (1984). Order Out of Chaos. Bantam. Bak, P., Tang, C. & Wiesenfeld, K. (1987). Self-organized criticality. Physical Review A , 38, 364–374. Strogatz, S. H. (1994). Nonlinear Dynamics and Chaos. Addison-Wesley. Kauffman, S. (1993). The Origins of Order. Oxford University Press.