Studying partial hyperbolicity inside regimes of motion in Hamiltonian systems

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.