Lambda-reciprocal iteration
Every step combines two contributions. First comes c·z(1−z), the quadratic logistic step behind the Lambda parameter set. Then comes a rational correction with fixed numerator 0.18 and denominator z² + c. A large denominator makes that correction small, so the map nearly follows the pure logistic version. Near z² + c = 0, it surges and may send the orbit far away.
Take c = 1 + i and start at z₀ = 0. The first term gives 1 × 0 × 1 = 0. The second term: 0.18 / (0 + (1+i)) = 0.18/(1+i) = 0.09 − 0.09i. So z₁ ≈ 0.09 − 0.09i: a small nudge away from the logistic baseline. On later steps the rational term keeps steering, creating paths unavailable to the pure logistic family. The denominator z² + c marks the poles—near-singular values—and their positions move with c.
