Integer Lur'e Workflow

Order-one describing-function and continuation route, anchored by the Chua q=1 reference case.

The integer Lur’e workflow runs the describing-function route at q=1q=1 for a system supplied in manual Lur’e form:

x˙=Ax+bψ(cTx).\dot{x}=Ax+b\psi(c^Tx).

It is intended for reusable order-one baselines and for checking a new system before adding fractional-memory effects.

Required System Data

  • vector field and parameters;
  • explicit AA, bb, cc and ψ(cTx)\psi(c^Tx);
  • classical describing function N(a)N(a);
  • equilibria for hiddenness controls;
  • preferably a Jacobian for Lyapunov calculations.

The library does not infer the Lur’e decomposition automatically.

Protocol

  1. Solve the Nyquist/describing-function closure for (ω0,k,a0)(\omega_0,k,a_0).
  2. Build the harmonic seed.
  3. Continue the residual nonlinearity from ε=0\varepsilon=0 to ε=1\varepsilon=1.
  4. Integrate the resulting target trajectory and compute dynamic diagnostics.
  5. Probe neighborhoods of each equilibrium and record whether any trajectory reaches the target.

Reference Case

The implemented benchmark is the non-smooth Chua system, with a linear-by-pieces characteristic, at q=1q=1. The regenerated corrected run records ω0=2.03918694\omega_0=2.03918694, k=0.20986735k=0.20986735, a0=5.85614509a_0=5.85614509, a positive leading Lyapunov exponent λ1=0.20824618\lambda_1=0.20824618, and 504 equilibrium-neighborhood probes with no TARGET hits.

See Chua Integer q=1 Reference for the model, figures, and evidence sources.

Python Entry Point

from hidden_attractors import get_system
from hidden_attractors.workflows.integer_lure import (
    continue_integer_lure_seed,
    final_integer_lure_attractor,
    integer_lure_seed,
    run_integer_lure_hiddenness_controls,
)

system = get_system("chua-nonsmooth")
seed = integer_lure_seed(system)
steps = continue_integer_lure_seed(system, seed)
target_seed, trajectory, status = final_integer_lure_attractor(
    system, steps[-1].x_out
)
probes = run_integer_lure_hiddenness_controls(system, trajectory)

For the complete historical Chua artifact family, use the compatibility command documented by the package:

hidden-attractors-integer-chua --help