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≥0. 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=√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.
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BibTeX
@misc{khaliliamirabadi2026etm3d,
title = {{ETM-3D: A One-Parameter Multistable Quadratic Flow with an Explicit Absorbing Set and Intermittent Chaos}},
author = {Roya Khalili-Amirabadi and Mohsen Jalaeian-Farimani},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04768},
eprint = {2610.04768},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04768},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - ETM-3D: A One-Parameter Multistable Quadratic Flow with an Explicit Absorbing Set and Intermittent Chaos
AU - Roya Khalili-Amirabadi
AU - Mohsen Jalaeian-Farimani
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04768
UR - https://arxiv.org/abs/2610.04768
AB - We study a one-parameter three-dimensional quadratic flow, denoted ETM-3D, on the benchmark domain ρ≥0. 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=√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.
ER -