A cyclically symmetric three-dimensional chaotic flow with saturation–oscillation coupling

We introduce a four-parameter autonomous three-dimensional system of ordinary differential equations (the Kornev system) whose only nonlinearity is the product of a sigmoidal saturation, tanh(αy), and a periodic modulation, cos(βz), arranged in a cyclic feedback loop with linear damping γ and linear cross-coupling δ: dx/dt = tanh(αy)cos(βz) − γx + δzdy/dt = tanh(αz)cos(βx) − γy + δxdz/dt = tanh(αx)cos(βy) − γz + δy The flow is equivariant under a group of order six, has constant divergence −3γ, and is proved to be dissipative with an explicit absorbing ball whenever γ > δ ≥ 0. At (α, β, γ, δ) = (2.5, 3.8, 0.19, 0.15) the system possesses a self-excited chaotic attractor with Lyapunov spectrum (0.505, 0, −1.075), Kaplan–Yorke dimension ≈ 2.47, correlation dimension ≈ 2.40 and 0–1 test statistic K ≈ 1.00. The phase space contains more than two thousand equilibria, eight of them stable, and a coexisting stable limit cycle. Bifurcation diagrams along all four parameters, basins of attraction and long-time statistics of transient chaos are reported. Files: manuscript (PDF) and Python code (NumPy, SciPy, Numba, Matplotlib) reproducing all results and figures.