A chaotic trajectory in weakly chaotic higher-dimensional Hamiltonian systems may locally present distinct regimes of motion, namely, chaotic, semiordered, or ordered. Such regimes, which are consequences of dynamical traps, are defined by the values of the Finite-Time Lyapunov Exponents (FTLEs) calculated during specific time windows. The Covariant Lyapunov Vectors (CLVs) contain the information about the local geometrical structure of the manifolds, and the distribution of the angles between them has been used to quantify deviations from hyperbolicity. In this work, we propose to study the deviation of partial hyperbolicity using the distribution of the local and mean angles during each of the mentioned regimes of motion. A system composed of two coupled standard maps and the Hénon–Heiles system are used as examples. Both are paradigmatic models to study the dynamics of mixed phase-space of conservative systems in discrete and continuous dynamical systems, respectively. Hyperbolic orthogonality is a general tendency in strong chaotic regimes. However, this is not true anymore for weakly chaotic systems and we must look separately at the regimes of motion. Furthermore, the distribution of angles between the manifolds in a given regime of motion allows us to obtain geometrical information about manifold structures in the tangent spaces. The description proposed here helps to explain important characteristics between invariant manifolds that occur inside the regimes of motion and furnishes a kind of visualization tool to perceive what happens in phase and tangent space of dynamical systems. This is crucial for higher-dimensional systems and to discuss distinct degrees of (non)hyperbolicity.

弱混沌高维哈密顿系统中的混沌轨迹可能会局部呈现不同的运动状态，即混沌，半有序或有序。这种机制是动态陷阱的结果，由在特定时间窗口内计算的有限时间李雅普诺夫指数（FTLE）的值定义。协变Lyapunov向量（CLV）包含有关歧管的局部几何结构的信息，并且它们之间的角度分布已用于量化与双曲线的偏差。在这项工作中，我们建议使用每个提到的运动方式期间局部和平均角度的分布来研究部分双曲率的偏差。以两个耦合的标准图和Hénon-Heiles系统组成的系统为例。两者都是范例模型，分别用于研究离散动力系统和连续动力系统中保守系统混合相空间的动力学。在强混沌状态下，双曲正交性是一个普遍的趋势。但是，对于弱混沌系统而言，这不再是正确的，我们必须分别研究运动状态。此外，在给定的运动状态下，歧管之间的角度分布使我们可以获得切线空间中歧管结构的几何信息。这里提出的描述有助于解释在运动状态内部发生的不变流形之间的重要特征，并提供一种可视化工具来感知动力系统的相空间和切线空间中发生的情况。