A Monotonicity Obstruction to Bounded Hyperchaos in an Integral-Feedback Polynomial Flow

A hyperchaotic attractor requires a bounded invariant set supporting at least two positive asymptotic Lyapunov exponents. We prove a global monotonicity obstruction for a four-dimensional integral-feedback extension of a polynomial chaotic flow. For , the transformation v = cw and effective gain reduce the two feedback parameters to one. When , satisfies . An omega-limit-set argument and an explicit calculation of the largest invariant subset of show that, for b > 0, every bounded forward trajectory converges to one of two equilibria. Hence, the nonzero-gain system admits no bounded periodic, quasiperiodic, chaotic, or hyperchaotic attractor. The zero-product cases are treated separately. For , the transformed variable v is constant; for c = 0, the base flow decouples from w. In both cases, a block-triangular variational argument shows that bounded non-equilibrium recurrent dynamics has at most one positive Lyapunov exponent. Thus, bounded hyperchaos is excluded for all . A theorem-guided numerical study combines a 2,806-node effective-gain atlas, 1,922 original-coordinate calculations, step-refined threshold events, and finite-range passage-time fits. For the three-dimensional base flow, a multistep differential-transform implementation is cross-checked against DOP853, while tangent-space QR calculations satisfy the exact divergence-sum identity. The combined results distinguish long irregular transients that mimic hyperchaos from bounded recurrent dynamics, which the monotonicity theorem excludes for every choice of the feedback parameters.