发表机构
Ferdowsi University of Mashhad; Politecnico di Milano（马什哈德 Ferdowsi 大学; 米兰理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究单参数二次流ETM-3D，通过李雅普诺夫函数证明全局有界性，分析分岔与间歇性，并区分于经典混沌族。
AI 中文摘要
我们研究了基准域ρ≥0上的单参数三维二次流，记为ETM-3D。其齐次二次三元组保持欧几里得二次能量，而完整向量场具有常数散度-8。一个平移的二次李雅普诺夫函数产生了一个显式的正不变吸收集，从而保证了全局最终有界性。平衡点一般可归结为一个标量四次方程，在ρ=√5处存在一个单独的奇异情形。延拓揭示了两个鞍结分岔、一个一般的跨临界分岔、一个超临界霍普夫分岔以及极限环的折叠。稳定和不稳定的周期一分支在折叠处相遇，其非平凡Floquet乘子趋近于+1。就在这一点之后，一个25点的对数参数研究给出了层流长度标度指数-0.551（95%置信区间[-0.621,-0.481]），与I型间歇性一致。在较大的ρ处，混沌吸引子与渐近稳定平衡共存，并出现周期窗口。结构不变量被用来将该流与几个经典二次混沌族区分开来。
英文摘要
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.
