发表机构
University of Zielona Góra; Universitat Autònoma de Barcelona; Reial Acadèmia de Ciències i Arts de Barcelona（热舒夫格奥拉大学; 巴塞罗那自治大学; 巴塞罗那皇家科学与艺术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出三角Nosé-Hoover振荡器，用有界三角函数替代二次项，研究其混沌、周期轨道与可积性，发现全局动力学与经典模型显著不同。
AI 中文摘要
我们引入了一种三角形式的Nosé-Hoover振荡器，其中二次力学项和无界恒温器耦合被有界三角函数所替代。该表述将谐波势替换为摆型势，并将恒温器相互作用限制为有界周期形式。由此产生的双参数系统自然定义在三维环面上，并在原点附近，至首阶，退化为经典多项式Nosé-Hoover模型。我们使用庞加莱截面、分岔图、李雅普诺夫谱、Kaplan-Yorke维数和李雅普诺夫可积性检验（LIT）来研究其全局动力学。数值结果揭示了规则动力学与混沌动力学的共存，并刻画了参数平面上耗散行为的变化。随后，我们分析了与参数轴相关的两个极限情形。对于$a=0$，我们在正则域上构造了两个函数独立的第一积分；而对于$b=0$，动力学简化为不变环面上的二维系统族，并使用达布多项式和指数因子对其进行分析。在这些可积极限交点附近的一阶平均产生了从未扰动周期轨道分岔的周期解，以及在其邻域内对正则$C^1$第一积分的阻碍。独立地，应用于正规变分方程的微分伽罗瓦理论，连同Ayoul-Zung和Li-Shi准则，排除了在$ab\ eq0$时特定非平衡相曲线附近的亚纯$B$-可积性和非常数亚纯第一积分。因此，尽管保留了经典Nosé-Hoover振荡器的局部结构，其三角对应物展现出显著不同的全局动力学和可积性。
英文摘要
We introduce a trigonometric version of the Nosé-Hoover oscillator in which the quadratic mechanical terms and unbounded thermostat coupling are replaced by bounded trigonometric functions. This formulation replaces the harmonic potential by a pendulum-type potential and confines the thermostat interaction to a bounded periodic form. The resulting two-parameter system is naturally defined on the three-dimensional torus and reduces near the origin, to leading order, to the classical polynomial Nosé-Hoover model. We investigate its global dynamics using Poincaré sections, bifurcation diagrams, Lyapunov spectra, Kaplan-Yorke dimensions, and the Lyapunov Integrability Test (LIT). The numerical results reveal the coexistence of regular and chaotic dynamics and characterize changes in dissipative behavior across the parameter plane. We then analyze the two limiting cases associated with the parameter axes. For $a=0$, we construct two functionally independent first integrals on regular domains, whereas for $b=0$ the dynamics reduces to a family of two-dimensional systems on invariant tori, which are analyzed using Darboux polynomials and exponential factors. First-order averaging near the intersection of these integrable limits yields periodic solutions bifurcating from unperturbed periodic orbits and an obstruction to regular $C^1$ first integrals in their neighborhoods. Independently, differential Galois theory applied to the normal variational equation, together with the Ayoul-Zung and Li-Shi criteria, excludes meromorphic $B$-integrability and non-constant meromorphic first integrals near a particular non-equilibrium phase curve for $ab\neq0$. Thus, despite retaining the local structure of the classical Nosé-Hoover oscillator, its trigonometric counterpart exhibits markedly different global dynamics and integrability.
Comments36 pages, 19 figures
