- Active matter
- Systems made of many self-propelled units that each consume energy to move, so the collective sits out of thermal equilibrium and can do things passive matter cannot — flock, swarm, or clump with no attraction at all. see Active Matter, Vicsek, Boids
- Associative memory
- A memory addressed by its content, not a location: show it a fragment or a corrupted copy of a stored pattern and it completes and cleans it back to the nearest whole memory. In an attractor network the stored patterns are the attractors at the bottoms of an energy landscape, and recall is just rolling downhill. see The Hopfield Network
- Attractor
- A state, or set of states, that a dynamical system tends toward and stays near — the place its behavior "settles." A fixed point, a periodic cycle, or a stranger object can each be an attractor. see Kuramoto, Game of Life (still lifes & oscillators)
- Bounded confidence
- The rule that people sway one another only when their views are already close enough; past a threshold of disagreement they simply ignore each other. How wide that threshold is decides whether a population converges to one consensus, splits into two camps, or shatters into many isolated fragments. see Opinion Dynamics
- Cellular automaton
- A grid of cells, each holding a state, all updated together in discrete time steps by the same local rule that looks only at a cell's neighbors. Simple to state, capable of astonishing behavior. see Game of Life, Elementary CA, Langton's Ant
- Chimera state
- A state of identical, symmetrically-coupled oscillators in which one part of the population locks into synchrony while the rest stays incoherent — coexisting order and disorder where the symmetry of the setup says there should be none. A striking form of symmetry breaking. see Chimera States
- Coarsening
- The slow growth of domains of like state — small patches merging into ever-larger ones — as a system relaxes toward uniformity. Its speed and the smoothness of the domain walls depend on whether the dynamics have surface tension. see Voter Model, Ising
- Computational irreducibility
- The property that some processes have no shortcut: the only way to find out what they do is to run them step by step. You cannot leap ahead to the answer, however clever your mathematics. see Langton's Ant, Elementary CA, The Edge of Chaos
- Complex contagion
- Spread that needs reinforcement: a node adopts only once a sufficient fraction of its neighbours already have — unlike simple contagion (a disease, a fire), where a single contact can suffice. Requiring a fraction makes whether something spreads depend on network density in surprising ways (see the cascade window). see Global Cascades; contrast The Forest-Fire Model
- Critical point / criticality
- The precise parameter value at a continuous phase transition. Exactly there, a system has no characteristic scale: fluctuations and correlations reach across all sizes at once, and the correlation length diverges. The most interesting place a system can sit. see Ising, Percolation, The Edge of Chaos
- Cyclic dominance
- Competition arranged in a loop — A beats B beats C beats A — so that no competitor can win outright. In space, the endless chase curls into spirals and lets all the players coexist. see Spatial Rock-Paper-Scissors
- Edge of chaos
- The narrow, fertile band between frozen order and boiling chaos, where a system is neither too rigid to change nor too noisy to remember — and where complex, computation-like behavior tends to live. An influential idea, and a contested one. see The Edge of Chaos, Game of Life, Elementary CA
- Emergence
- The appearance of structure, pattern, or behavior at the scale of the whole that is present in none of the parts and was written into none of the local rules. The subject of this entire Atlas. see What Emergence Is
- Excitable medium
- A medium whose every point can be quiet, fire when sufficiently stimulated, and then must recover before it can fire again. That cycle lets waves propagate one-directionally and curl into spirals. see Excitable Media
- Fractal
- A shape that looks similar at every magnification and is rough enough that its effective ("fractal") dimension is not a whole number. Branching, crinkled, scale-free form. see Diffusion-Limited Aggregation, L-Systems, Percolation
- Limit cycle
- A stable, isolated, self-sustaining periodic orbit — a rhythm a system falls into and returns to after a nudge. The reason a metronome keeps a steady beat and a Lenia creature holds its shape. see Coupled Metronomes, Kuramoto, Lenia
- Order parameter
- A single number that measures how ordered a system is — typically running from 0 (disordered) to 1 (fully ordered). Watching it cross a threshold is how you spot a phase transition. see Kuramoto (r), Vicsek (vₐ), Ising (magnetization)
- Phase transition
- An abrupt, qualitative change in a system's collective state as a control parameter crosses a threshold — solid to liquid, disordered to ordered, free-flowing to jammed. see Ising, Vicsek, Phantom Traffic Jams, Percolation
- Percolation threshold
- The critical density at which scattered local connections suddenly link up into a single cluster that spans the whole system — the onset of long-range connectivity. see Percolation, The Forest-Fire Model
- Positive feedback
- A loop in which an effect amplifies its own cause, so tiny early differences compound into enormous ones — "the rich get richer." It drives runaway growth, tipping points, and winner-take-all outcomes, and it is the engine behind many of the Atlas's most lopsided results. see Preferential Attachment, Wealth Condensation, Global Cascades
- Power law
- A relationship in which the frequency of an event falls off as a power of its size (frequency ∝ size−α). Such distributions are heavy-tailed and scale-free: huge events are rare but not vanishingly so. see The Abelian Sandpile, The Forest-Fire Model, Preferential Attachment
- Reaction–diffusion / Turing pattern
- Pattern that forms when locally reacting substances also spread, with a slow, far-reaching inhibitor and a fast, short-range activator. Alan Turing's counter-intuitive result: diffusion, the great smoother, can create structure. see Reaction–Diffusion
- Refractory period
- The recovery time after a unit fires, during which it cannot fire again. It is what keeps excitation waves moving forward instead of backward, and what lets spirals persist. see Excitable Media, Fireflies
- Scale-free
- Having no characteristic size: the structure or its statistics look the same whatever scale you examine. The signature of fractals, power laws, and systems sitting at criticality. see Preferential Attachment, The Abelian Sandpile
- Self-organization
- The spontaneous appearance of order from local interactions, with no leader, blueprint, or outside hand arranging it. Emergence, seen from the angle of the order doing the arranging itself. see Boids, Slime Mould Networks, Reaction–Diffusion
- Self-organized criticality (SOC)
- The tendency of some slowly-driven systems to tune themselves to a critical state and stay there — producing avalanches of every size (power-law distributed) with nobody adjusting any dial. see The Abelian Sandpile, The Forest-Fire Model
- Stigmergy
- Coordination through traces left in a shared environment rather than through direct communication — an ant lays a scent another ant follows; the world itself carries the message. see Ant Colony Foraging, Slime Mould Networks
- Symmetry breaking
- When a system whose rules treat several outcomes as equivalent nonetheless settles on one of them — the magnet points up or down, one oscillator group leads — so the result carries less symmetry than the law that produced it. The chosen outcome is set by chance or tiny initial differences, and it is one of the deepest signatures of emergence. see The Ising Model, Chimera States
- Synchronization / phase locking
- The convergence of coupled oscillators onto a common rhythm — fireflies flashing together, metronomes ticking as one — with no conductor and no shared clock. see Kuramoto, Fireflies, Coupled Metronomes
- Turing-completeness / universal computation
- The capacity to compute anything that can be computed, given enough space and time. Astonishingly, some of the simplest rules in the Atlas have it. see Game of Life, Elementary CA (rule 110)
- Universality
- The remarkable fact that near a critical point, the microscopic details stop mattering: utterly different systems fall into a few shared classes with identical critical behavior. At the edge, nature forgets what it is made of. see Ising, The Edge of Chaos