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Attractor Analysis for the TEO Phase Space

An analysis programme; fixed points, cycles, chaos, and fractal basins must be established, not inferred from their names.

Status

TEO defines a coupled dynamical system, so questions about equilibria, stability, periodic orbits, and basins are well posed. The current repository simulations exhibit selected trajectories and parameter sweeps. They do not yet classify every attractor, prove a limit cycle, establish chaos, or measure a fractal basin boundary.

State and parameters

Let the model state be

\[ z=(x_1,\ldots,x_N,\theta_1,\ldots,\theta_N,\Omega,H). \]

For fixed parameters \(p\), write its dynamics as

\[ \dot z=F(z;p). \]

The first analysis task is to find admissible equilibria \(z^*\) satisfying

\[ F(z^*;p)=0 \]

under the simplex, phase, and substrate constraints.

Fixed points

Local stability of a differentiable equilibrium is assessed from the Jacobian

\[ J(z^*;p)=\frac{\partial F}{\partial z}(z^*;p). \]

If all relevant eigenvalues have negative real parts after accounting for neutral phase symmetries and constraints, the equilibrium is locally asymptotically stable.

Positive regulation, coupling above a Kuramoto threshold, and bounded overshoot do not by themselves prove convergence to a unique equitable fixed point. That conclusion depends on the fitness functions, graph, frequency distribution, brake, boundary conditions, and initial state.

Periodic and chaotic behaviour

A periodic trace is not enough to establish a stable limit cycle. Evidence should include a closed orbit or Poincaré section, attraction from nearby initial states, and numerical convergence checks.

Likewise, irregular output is not enough to establish chaos. A chaos claim should report, at minimum:

  • a positive maximal Lyapunov exponent with convergence diagnostics;
  • sensitivity across nearby initial states;
  • bounded aperiodic trajectories over adequate horizons;
  • robustness to integration step and solver choice;
  • exclusion of transient or numerical artefacts.

No general “edge of chaos” intelligence conclusion follows from a positive exponent.

Basins and the viable set

A basin of attraction is defined only after an attractor is identified. Mapping a basin requires sampling initial states from a declared measure and classifying their long-run destinations with explicit tolerances.

The viable corridor is instead an admissible region defined by selected constraint conditions. A viable set need not be an attractor basin, and a basin boundary need not be fractal. Those are separate properties to test.

Network topology

For an undirected communication graph, \(\lambda_2>0\) establishes connectivity. It does not establish structural resilience under node loss. A Chord-style state would need its own commit-time composition test; it is not implied by the attractor type or Fiedler value.

Concrete next experiment

  1. choose a small \(N\) and freeze all model functions and parameters;
  2. solve or numerically locate equilibria;
  3. compute constrained Jacobian spectra;
  4. continue equilibria across \(\gamma\), \(K\), and throughput parameters;
  5. test for bifurcations;
  6. sample initial conditions and estimate basin volumes;
  7. only then test periodicity, chaos, or boundary geometry.

The output should distinguish theorem, numerical observation, and solver-dependent diagnostic.

Current conclusion

Attractor language is appropriate for TEO, but the original three-way classification was a hypothesis presented as a result. What survives is a clear mathematical work programme.