Global dynamics visualisation from Lagrangian Descriptors. Applications to discrete and continuous systems
Jerome Daquin, Remi Pedenon-Orlanducci, Makrina Agaoglou, Guillermo Garcia-Sanchez, Ana Maria Mancho
备注：
v3: Accepted for publication in Physica D. Minor modifications. 4 references added. 22 pages, 64 references. v2: Slight modification of the title. 15 references added, 2 Figures added. v1: 22 pages, 9 figures, 60 references. Comments and feedback are welcome
This paper introduces a new global dynamics and chaos indicator based on the method of Lagrangian Descriptor apt for discriminating ordered and deterministic chaotic motions in multidimensional systems. The selected implementation of this method requires only the knowledge of orbits on finite time windows and is free of the computation of the tangent vector dynamics (i.e., variational equations are not needed). To demonstrate its ability in visualising different dynamical behaviors, in particular for highlighting chaotic regions, several stability maps of classical systems, obtained with different phase space methods, are reproduced. The benchmark examples are rooted in discrete and continuous nearly-integrable dynamical systems, with prominent features played by resonances. These include the Chirikov standard map, higher dimensional symplectic and volume preserving maps, fundamental models of resonances, and a 3 degrees-of-freedom nearly-integrable Hamiltonian system with a dense web of resonances. The indicator thus appears to be relevant for understanding phase space transport mediated by resonances in nearly-integrable system, as ubiquitous in celestial mechanics or astrodynamics.
Using Lagrangian descriptors to reveal the phase space structure of dynamical systems described by fractional differential equations: Application to the Duffing oscillator
An Extension of Discrete Lagrangian Descriptors for Unbounded Maps
本文将离散拉格朗日描述子方法推广到无界映射的相空间中。关键思想是构建一个工作定义，该定义建立在Lopesino et al.（2015）中引入的原始方法的基础上，并且依赖于当它们的轨道离开平面中的某个区域时停止初始条件的迭代。这个准则部分地受到动力系统理论中使用的经典分析的启发，该分析通过逃逸时间图来研究映射的动力学。我们说明了这种技术的能力，以揭示相空间中的稳定和不稳定的不变流形的几何模板，以及复杂的结构的混沌集和奇异吸引子，通过应用它来揭开一个著名的离散时间系统，Hénon映射的相空间。
Comparing Finite-Time Lyapunov Exponents and Lagrangian Descriptors for identifying phase space structures in a simple two-dimensional, time-periodic double-gyre model
Phase space integrity in neural network models of Hamiltonian dynamics: A Lagrangian descriptor approach
Department of Mathematics, Khalifa University of Science and Technology（哈利法科学技术大学数学系）；Public Health and Epidemeology Department, Khalifa University of Science and Technology（哈利法科学技术大学公共卫生与流行病学系）
