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Attractor Geometry of the TEO Phase Space

Classifying the dynamical regimes of intelligent collectives through attractor topology.

Motivation

The TEO system couples replicator dynamics, Kuramoto synchronization, and a thermodynamic constraint into a single phase space. The long-term behavior of this coupled system is governed by its attractors — the states toward which trajectories converge. Understanding the attractor geometry tells us what kinds of stable configurations (civilizations, agent ecologies, organizations) the system can produce.

The Three Attractor Types in TEO

1. Fixed Point Attractors — Stable Equilibria

When homeostasis (\(\gamma > 0\)), cultural coupling (\(K > K_c\)), and entropy budget (\(dS/dt < D_{\max}\)) are all satisfied, the TEO system converges to a fixed point:

  • Resource shares \(x_i\) stabilize around \(x_i \approx 1/N\) (equitable distribution)
  • Value orientations \(\theta_i\) lock to a common phase \(\psi\) (consensus)
  • Entropy production settles below \(D_{\max}\) (sustainability)

This is the Chord equilibrium: a stable, high-IP state where all governance constraints operate simultaneously. The basin of attraction volume measures how many initial conditions lead to this outcome — it shrinks as \(\gamma\) decreases or \(K\) drops below \(K_c\).

2. Limit Cycle Attractors — Oscillatory Regimes

Near the Kuramoto critical coupling \(K \approx K_c\), the system can enter limit cycles: periodic oscillations between partial synchronization and desynchronization. In civilizational terms, these correspond to recurring cycles of consensus and polarization.

The Kuramoto order parameter oscillates: \(r(t) = r_0 + \epsilon \sin(\omega t)\). Resource allocation follows, as agents aligned with the momentary consensus gain fitness advantages that reverse when the phase shifts.

These cycles are structurally stable — perturbations shift the phase but not the cycle. They represent a society (or agent ecology) that neither fully polarizes nor fully synchronizes, but perpetually oscillates between the two.

3. Chaotic Attractors — Strange Attractors at the Edge

When the system operates near multiple simultaneous critical thresholds — \(K \approx K_c\), \(\gamma \approx 0\), and \(dS/dt \approx D_{\max}\) — the TEO dynamics become chaotic: trajectories are bounded but aperiodic, with sensitive dependence on initial conditions.

The maximum Lyapunov exponent \(\lambda_{\max}\) characterizes this regime:

\[\lambda_{\max} = \lim_{t \to \infty} \frac{1}{t} \ln \frac{|\delta \mathbf{z}(t)|}{|\delta \mathbf{z}(0)|}\]

where \(\mathbf{z} = (x_1, \ldots, x_N, \theta_1, \ldots, \theta_N)\) is the full state vector.

  • \(\lambda_{\max} < 0\): fixed point (stable)
  • \(\lambda_{\max} = 0\): limit cycle (neutral stability)
  • \(\lambda_{\max} > 0\): chaos (exponential divergence of nearby trajectories)

This is the Edge of Chaos regime — and, per Claim 8 of the Emergence Manifesto, potentially where maximal information processing occurs.

Basin of Attraction Structure

The TEO phase space is partitioned into basins of attraction. Each basin maps a set of initial conditions to a specific long-term behavior. The basin boundary is a fractal in chaotic regimes — meaning that arbitrarily small differences in initial conditions can lead to qualitatively different outcomes.

For the teo-civilization simulation, the key parameter axes are:

Parameter Low High
\(\gamma\) (homeostasis) Monopoly basin Equity basin
\(K\) (coupling) Polarization basin Consensus basin
\(D_{\max}\) (entropy budget) Collapse basin Sustainability basin

The intersection of all three "high" basins — equity, consensus, sustainability — is the viable corridor. The TEO simulation demonstrates that this corridor exists but is narrow.

Connection to Identity Persistence

In the Chord equilibrium (fixed point attractor), all governance constraints co-exist simultaneously: the agent ecology maintains structural resilience (\(\lambda_2 > 0\)), thermodynamic sustainability (\(dS/dt < D_{\max}\)), and cognitive persistence (\(\text{IP} \to 1\)).

In the Arpeggio regime (limit cycle or chaotic attractor), identity components flicker in and out of the operative set, producing the time-multiplexed, unstable identity described in lerchner-boundary.md.

The attractor type thus determines whether a system can achieve the Chord state — it is a topological precondition, not merely a parameter choice.