ETM-3D: A One-Parameter Multistable Quadratic Flow with an Explicit Absorbing Set and Intermittent Chaos

Abstract

We study a one-parameter three-dimensional quadratic flow, denoted ETM-3D, on the benchmark domain $ρ\geq0$. Its homogeneous quadratic triad preserves Euclidean quadratic energy, while the full vector field has constant divergence -8. A shifted quadratic Lyapunov function yields an explicit positively invariant absorbing set and therefore global ultimate boundedness. The equilibria reduce generically to a scalar quartic, with a separate singular case at $ρ=\sqrt{5}$. Continuation reveals two saddle-node bifurcations, a generic transcritical bifurcation, a supercritical Hopf bifurcation, and a fold of limit cycles. Stable and unstable period-one branches meet at the fold with their nontrivial Floquet multipliers approaching +1. Just beyond this point, a 25-point logarithmic parameter study gives a laminar-length scaling exponent of -0.551 (95% confidence interval [-0.621,-0.481]), consistent with type-I intermittency. At larger $ρ$, a chaotic attractor coexists with an asymptotically stable equilibrium and periodic windows occur. Structural invariants are used to distinguish the flow from several classical quadratic chaotic families.