In this paper we apply the method of Lagrangian descriptors to explore the geometrical structures in phase space that govern the dynamics of dissipative systems. We demonstrate through many classical examples taken from the nonlinear dynamics literature that this mathematical technique can provide valuable information and insights to develop a more general and detailed understanding of the global behavior and underlying geometry of these systems. In order to achieve this goal, we analyze systems that display dynamical features such as hyperbolic points with different expansion and contraction rates, limit cycles, slow manifolds and strange attractors. Furthermore, we study how this technique can be used to detect transition ellipsoids that arise in Hamiltonian systems subject to dissipative forces, and which play a crucial role in characterizing trajectories that evolve across an index-1 saddle point of the underlying potential energy surface.

在本文中，我们应用拉格朗日描述符的方法来探索相空间中控制耗散系统动力学的几何结构。我们通过许多取自非线性动力学文献的经典例子证明，这种数学技术可以提供有价值的信息和见解，以更全面和详细地了解这些系统的全局行为和基础几何结构。为了实现这一目标，我们分析了显示动态特征的系统，例如具有不同膨胀和收缩率的双曲线点、极限环、慢流形和奇怪的吸引子。此外，我们研究了如何使用这种技术来检测受耗散力影响的哈密顿系统中出现的过渡椭球，