The Rössler System with a State-Dependent Delay: Hopf Criticality, Chaos and Key Recovery

We study the Rössler system with a delayed feedback 𝑘⁢𝑧⁢(𝑡 −𝑡⁡(𝑧)) in the 𝑧-equation and state-dependent delay 𝑡⁡(𝑧) =𝑡0 +𝑐1⁢𝑧2. The linearization coincides with that of a constant delay 𝑡∗ =𝑡0 +𝑐1⁢𝑧∗2, so the Hopf curves in the (𝑘,𝑡0) plane are shifted exactly by 𝑐1⁢𝑧∗2 (verified by a pseudospectral method and by direct simulation). In a regime that possesses a stable equilibrium, the state dependence alone switches the Hopf bifurcation from subcritical to supercritical: the first Lyapunov coefficient computed with the normal-form theory for SD-DDEs changes sign at 𝑐∗ 1 ≈0.389, and continuation of periodic orbits (DDE-BIFTOOL) confirms the picture, including the folds of the subcritical branches. In the chaotic regime, an ensemble of 16 initial conditions with six Lyapunov exponents shows a small but statistically detectable decrease of the chaotic fraction with 𝑐1 (from 0.53 to 0.48), while individual parameter points are reclassified at a rate comparable to the initial-condition variability; no hyperchaos is found. When 𝑐1 is regarded as a chaos-based key it produces logarithmic sensitivity, yet a least-squares attack on the observed signal 𝑧⁡(𝑡) recovers (𝑡0,𝑐1) even when 𝑎,𝑏,𝑐,𝑘 are also unknown and the channel adds noise, band-limiting and quantization; the scheme is therefore not secure by itself.