A general formalism describing a type of energy-conservative system is established. Some possible dynamic behaviors of such energy-conservative systems are analyzed from the perspective of geometric invariance. A specific 4D chaotic energy-conservative system with a line of equilibria is constructed and analyzed. Typically, an energy-conservative system is also conservative in preserving its phase volume. The constructed system however is conservative only in energy but is dissipative in phase volume. It produces energy-conservative attractors specifically exhibiting chaotic 2-torus and quasiperiodic behaviors including regular 2-torus and 3-torus. From the basin of attraction containing a line of equilibria, the hidden nature of chaotic attractors generated from the system is further discussed. The energy hypersurface on which the attractors lie is determined by the initial value, which generates complex dynamics and multistability, verified by energy-related bifurcation diagrams and Poincaré sections. A new type of coexistence of attractors on the equal-energy hypersurface is discovered by turning the system parameter values to their opposite. The basins of attraction under three sets of parameter values demonstrate that the Hamiltonian is the leading factor predominating the dynamic behaviors of the system with a closed energy hypersurface. Finally, an analog circuit is designed and implemented to demonstrate the consistent theoretical and simulation results.

中文翻译：

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全局不变超曲面上的稀有能量守恒吸引子及其多重稳定性

建立了描述一种能量守恒系统的一般形式。从几何不变性的角度分析了这种能量守恒系统的一些可能的动态行为。构建并分析了具有平衡线的特定4D混沌能量守恒系统。通常，能量守恒系统在保持其相体积方面也是保守的。然而，所构建的系统仅在能量上是保守的，但在相体积上是耗散的。它产生能量守恒吸引子，特别表现出混沌 2-环面和准周期行为，包括常规 2-环面和 3-环面。从包含一条平衡线的吸引盆中，进一步讨论了系统产生的混沌吸引子的隐藏性质。吸引子所在的能量超曲面由初始值确定，该初始值会产生复杂的动力学和多稳定性，并通过与能量相关的分岔图和庞加莱截面进行验证。通过将系统参数值变为相反，发现了一种新型的等能超曲面上的吸引子共存。三组参数值下的吸引力盆地表明，哈密顿量是主导具有闭合能量超曲面的系统动态行为的主导因素。最后，设计并实现了一个模拟电路，以展示一致的理论和仿真结果。通过将系统参数值变为相反，发现了一种新型的等能超曲面上的吸引子共存。三组参数值下的吸引力盆地表明，哈密顿量是主导具有闭合能量超曲面的系统动态行为的主导因素。最后，设计并实现了一个模拟电路，以展示一致的理论和仿真结果。通过将系统参数值变为相反，发现了一种新型的等能超曲面上的吸引子共存。三组参数值下的吸引力盆地表明，哈密顿量是主导具有闭合能量超曲面的系统动态行为的主导因素。最后，设计并实现了一个模拟电路，以展示一致的理论和仿真结果。