arXiv subjects

V. J. García-Garrido

Publications and source records attributed to V. J. García-Garrido.

10 recordsLinked to original sources

arXiv2026

Quantization of Lagrangian Descriptors

We formulate Lagrangian descriptors (LDs) in the path integral framework. Averaging the classical LD over fluctuations about extremal trajectories defines a quantum LD that incorporates quantum effects. Invariant manifolds, which sharply organize classical transport, become finite-width phase space structures under quantum fluctuations, and their overlap provides a geometric mechanism consistent with tunneling as fluctuation-induced delocalization of transport barriers. We demonstrate this approach for the Hamiltonian saddle, where path integral sampling reveals manifold broadening and barrier penetration. This establishes a geometric framework for studying phase space transport and tunneling beyond the classical regime, while also providing a natural route toward the application of LDs to field theory.

math.DS↗

arXiv2026

Trajectory arclength reveals chaos

In this paper we demonstrate that the phase space arclength of a trajectory, quantified by the time-averaged Lagrangian descriptor, is a robust and self-contained chaos indicator. By invoking Birkhoff's Ergodic Partition Theorem, we show that this scalar function distinguishes dynamical regimes via its power spectral density: for regular motion it converges to a delta function, whereas for chaotic trajectories the spectrum exhibits an inverse power-law $(1/ω)$ driven by the phenomenon of dynamical stickiness. With this approach, we avoid the computation and simulation of the variational equations and the usage of neighboring orbits, making it the simplest geometrical chaos indicator derivable from Lagrangian descriptors. Its computational efficiency enables the study of high-dimensional systems and allows the generation of large datasets of classified initial conditions, ideal for training Machine Learning models. We validate these findings using the Hénon-Heiles and the Fermi-Pasta-Ulam systems. By linking the geometrical properties of phase space to spectral analysis, this work provides the mathematical justification to establish Lagrangian descriptors as a rigorous, self-sufficient framework for the global analysis of chaos and regularity in Hamiltonian systems.

nlin.CD↗

arXiv2024

Chaos and Regularity in the Double Pendulum with Lagrangian Descriptors

In this paper we apply the method of Lagrangian descriptors as an indicator to study the chaotic and regular behavior of trajectories in the phase space of the classical double pendulum system. In order to successfully quantify the degree of chaos with this tool, we first derive Hamilton's equations of motion for the problem in non-dimensional form, showing that they can be written compactly using matrix algebra. Once the dynamical equations are obtained, we carry out a parametric study in terms of the system's total energy and the other model parameters (lengths and masses of the pendulums, and gravity), to determine the extent of the chaotic and regular regions in the phase space. Our numerical results show that for a given mass ratio, the maximum chaotic fraction of phase space trajectories is attained when the pendulums have equal lengths. Moreover, we give a characterization of the growth and decay of chaos in the system in terms of the model parameters, and explore the hypothesis that the chaotic fraction follows an exponential law over different energy regimes.

math.DS↗

arXiv2022

Lagrangian Descriptors and the Action Integral of Classical Mechanics

In this paper we bring together the method of Lagrangian descriptors and the principle of least action, or more precisely, of stationary action, in both deterministic and stochastic settings. In particular, we show how the action can be used as a Lagrangian descriptor. This provides a direct connection between Lagrangian descriptors and Hamiltonian mechanics, and we illustrate this connection with benchmark examples.

math.DS↗

arXiv2021

The Dynamical Significance of Valley-Ridge Inflection Points

In this paper we demonstrate that valley-ridge inflection (VRI) points of a potential energy surface (PES) have a dynamical influence on the fate of trajectories of the underlying Hamiltonian system. These points have attracted the attention of chemists in the past decades when studying selectivity problems in organic chemical reactions whose energy landscape exhibits a post-transition-state bifurcation in the region between two sequential saddles without an intervening energy minimum. To address the dynamical significance of valley-ridge inflection points, we construct a symmetric potential energy function that allows us to move the location of the VRI point while keeping the locations and energies of the critical points fixed. In this setup, we carry out a parametric study of the dynamical behavior of ensembles of trajectories in terms of the energy of the chemical system and the position of the VRI point. Our analysis reveals that the location of the VRI point controls the fraction of trajectories that recross the high energy saddle region of the PES without entering either of the potential wells that are separated by the low energy saddle.

math.DS↗

arXiv2021

Painting the Phase Space of Dissipative Systems with Lagrangian Descriptors

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 tool 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.

math.DS↗

arXiv2020

Phase Space Analysis of the Dynamics on a Potential Energy Surface with an Entrance Channel and Two Potential Wells

In this paper we unveil the geometrical template of phase space structures that governs transport in a Hamiltonian system described by a potential energy surface with an entrance/exit channel and two wells separated by an index-1 saddle. For the analysis of the nonlinear dynamics mechanisms, we apply the method of Lagrangian descriptors, a trajectory-based scalar diagnostic tool that is capable of providing a detailed phase space tomography of the interplay between the invariant manifolds of the system. Our analysis reveals that, the stable and unstable manifolds of two families of unstable periodic orbits (UPOs) that exist in the regions of the wells are responsible for controlling the access to the wells of trajectories that enter the system through the channel. In fact, we demonstrate that the heteroclinic and homoclinic connections that arise in the system between the manifolds of the families of UPOs characterize the branching ratio, a relevant quantity used to measure product distributions in chemical reaction dynamics.

nlin.CD↗

arXiv2020

Tuning the Branching Ratio in a Symmetric Potential Energy Surface with a Post-Transition State Bifurcation using External Time Dependence

Chemical selectivity, as quantified by a branching ratio, is a phenomenon relevant for many organic chemical reactions. It may be exhibited on a potential energy surface that features a valley-ridge inflection point in the region between two sequential index-1 saddles, with one saddle having higher energy than the other. Reaction occurs when a trajectory crosses the region of the higher energy saddle (the entrance channel) and approaches the lower energy saddle. On both sides of the lower energy saddle, there are two wells and the question we address in this work is that, given an initial ensemble of trajectories, what is the relative fraction of trajectories that enter each well. For a symmetric PES this fraction is 1:1. We consider a symmetric PES subjected to a time-periodic forcing characterized by an amplitude, frequency, and phase. In this letter we analyse how the branching ratio depends on these three parameters.

physics.chem-ph↗

arXiv2019

The Dynamical Matching Mechanism in Phase Space for Caldera-Type Potential Energy Surfaces

Dynamical matching occurs in a variety of important organic chemical reactions. It is observed to be a result of a potential energy surface (PES) having specific geometric features. In particular, a region of relative flatness where entrance and exit to this region is controlled by index-one saddles. Examples of potential energy surfaces having these features are the so-called caldera potential energy surfaces. We develop a predictive level of understanding of the phenomenon of dynamical matching in a caldera potential energy surface. We show that the phase space structure that governs dynamical matching is a particular type of heteroclinic trajectory which gives rise to trapping of trajectories in the central region of the caldera PES. When the heteroclinic trajectory is broken, as a result of parameter variations, then dynamical matching occurs.

nlin.CD↗

arXiv2016

Response to: "Limitations of the Method of Lagrangian Descriptors" [arXiv:1510.04838]

This Response is concerned with the recent Comment of Ruiz-Herrera, "Limitations of the Method of Lagrangian Descriptors" [arXiv:1510.04838], criticising the method of Lagrangian Descriptors. In spite of the significant body of literature asserting the contrary, Ruiz-Herrera claims that the method fails to reveal the presence of stable and unstable manifolds of hyperbolic trajectories in incompressible systems and in almost all linear systems. He supports this claim by considering the method of Lagrangian descriptors applied to three specific examples. However in this response we show that Ruiz-Herrera does not understand the proper application and interpretation of the method and, when correctly applied, the method beautifully and unambiguously detects the stable and unstable manifolds of the hyperbolic trajectories in his examples.

math.DS↗