Mathematical and Dynamical Foundations › Fixed Points, Stability, and Attractors

Prediction alone does not characterize a dynamical system. Ask where trajectories end up if you let them run, and you are asking about fixed points, stability, and attractors. These concepts are not optional decoration for world modelers: they determine whether a learned model's long-horizon rollouts look like the real system or like garbage, and they explain why one-step validation can be dangerously misleading. The shift in perspective is from local to global: a transition law tells you what happens in the next instant, while attractor structure tells you what happens in the long run, and the two can come apart in ways that matter enormously.

Fixed points matter because they organize the long-term behavior of everything around them. They act as the destinations toward which nearby trajectories drift or the sources from which they flee, and the inventory of a system's fixed points is therefore a rough sketch of its entire long-run personality. A trained world model whose rollouts converge to a spurious fixed point, for example a video model that predicts the blurry mean of all plausible futures, has failed in a specifically dynamical way, even if each individual one-step prediction looked plausible. The failure is invisible to one-step metrics because the spurious fixed point is approached gradually, one defensible step at a time.

Not all fixed points are equal. Some pull nearby trajectories in; others fling them away. The distinction is captured by stability. Consider a one-dimensional discrete-time system $x_{t+1} = f(x_t)$ with fixed point $x^*$. Nudge the state slightly: $x_t = x^* + \delta_t$. A Taylor expansion gives

$$x^* + \delta_{t+1} \approx f(x^*) + f'(x^*)\,\delta_t = x^* + f'(x^*)\,\delta_t,$$

so the perturbation evolves as $\delta_{t+1} \approx f'(x^*)\,\delta_t$. The nudge shrinks if $|f'(x^*)| < 1$ and grows if $|f'(x^*)| > 1$. Linearization therefore sorts fixed points into stable (perturbations decay) and unstable (perturbations grow) ones. The intuition is geometric: the derivative measures how strongly the map stretches or compresses small intervals, and repeated application multiplies that stretch factor, so anything less than one in magnitude decays geometrically while anything greater explodes geometrically.

The same idea in higher dimensions replaces the derivative with the Jacobian matrix $J = \partial f / \partial s$ evaluated at the fixed point, and the condition becomes: all eigenvalues of $J$ must lie strictly inside the unit circle of the complex plane. Each eigenvalue governs perturbations along its own eigendirection, so a system can be stable along one direction and unstable along another, which is why the worst eigenvalue decides. For continuous-time systems $\dot{s} = g(s)$, the condition is that all eigenvalues of the Jacobian have negative real part. If you take one piece of classical control mathematics into the rest of this handbook, make it this one, because it is exactly the diagnostic you will run, knowingly or not, whenever a learned model's rollouts blow up, oscillate, or collapse. A learned dynamics network is a function like any other; it has Jacobians at its fixed points, and its eigenvalues decide whether its imagined futures settle down or spiral apart.

Linearization is a local tool. It tells you what happens in an infinitesimal neighborhood of the fixed point and says nothing about which initial conditions eventually arrive there. That second question brings us to basins and attractors.

An attractor is a set that trajectories converge to and stay near. The main species, in increasing order of complexity:

• Fixed-point attractors: convergence to a single state, like a cup of coffee cooling to room temperature. All memory of the initial condition is eventually lost.
• Limit cycles: convergence to a repeating orbit, like a heartbeat or, as we will see shortly, a thermostat. The system forgets when it started but remembers a rhythm.
• Quasi-periodic tori: motion on two or more incommensurate frequencies, common in coupled oscillators. The motion never exactly repeats, yet remains regular and predictable over moderate horizons.
• Strange attractors: bounded but aperiodic behavior with sensitive dependence on initial conditions, the signature of chaos. The system stays in a recognizable region forever yet becomes unpredictable in detail.

Each attractor comes with a basin of attraction: the set of initial conditions that converge to it. Systems with several attractors partition their state space into basins, and which basin a trajectory starts in can matter more than any local detail of its motion. Basins introduce a discrete, qualitative sensitivity into otherwise smooth dynamics: nudge an initial condition across a basin boundary and its destiny changes completely, even though its immediate next steps barely change. Learned world models routinely distort basin boundaries, which is one reason a model can behave well in the data distribution yet fail catastrophically for the states a planner steers it toward. A planner that believes a boundary sits one place, when it actually sits somewhere else, will confidently steer across it and discover a different regime than its model promised.

The cleanest laboratory for these ideas is the logistic map,

$$x_{t+1} = r\,x_t (1 - x_t),$$

a one-dimensional, deterministic, nonlinear system studied extensively by Robert May in the 1970s as a model of a self-limited population. Despite its simplicity, as the parameter $r$ increases it exhibits a fixed point, a cascade of periodic oscillations, and finally chaos. One quadratic term, one parameter, and every phenomenon in the attractor taxonomy above: this economy is why the map remains the standard first example four decades on.

We can analyze it exactly. The fixed points are $x^* = 0$ and $x^* = (r-1)/r$. Since $f'(x) = r(1 - 2x)$, the derivative at the nonzero fixed point is $f'(x^*) = 2 - r$, which is stable precisely when $|2 - r| < 1$, that is, for $1 < r < 3$. At $r = 3$ the fixed point loses stability and a stable period-two cycle is born. That cycle itself loses stability at $r = 1 + \sqrt{6} \approx 3.45$, giving way to period four, then eight, in a period-doubling cascade with remarkably universal spacing (Feigenbaum's constant $\delta \approx 4.669$), until chaos sets in near $r \approx 3.57$. Each doubling is a bifurcation: a stable attractor splits in two, and the system's long-run behavior acquires one more alternating phase.