المؤلفون: Katsanikas, M., Agaoglou, M., Wiggins, S., Mancho, A. M.

مصطلحات موضوعية: Nonlinear Sciences - Chaotic Dynamics, Mathematical Physics, Mathematics - Dynamical Systems, Physics - Chemical Physics

الوصف: We apply the method of Lagrangian Descriptors (LDs) to a symmetric Caldera-type potential energy surface which has three index-1 saddles surrounding a relatively flat region that contains no minimum. Using this method we show the phase space transport mechanism that is responsible for the existence and non-existence of the phenomenon of dynamical matching for this form of Caldera potential energy surface.
Comment: 10 pages, 7 figures

الوصول الحر: http://arxiv.org/abs/2205.14770Test

المؤلفون: García-Garrido, V. J., Wiggins, S.

مصطلحات موضوعية: Mathematics - Dynamical Systems, Nonlinear Sciences - Chaotic Dynamics

الوصف: 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.

الوصول الحر: http://arxiv.org/abs/2204.04728Test

المؤلفون: Katsanikas, M., Wiggins, S.

مصطلحات موضوعية: Nonlinear Sciences - Chaotic Dynamics, Mathematical Physics, Mathematics - Dynamical Systems, Physics - Chemical Physics

الوصف: We present a method that generalizes the periodic orbit dividing surface construction for Hamiltonian systems with three or more degrees of freedom. We construct a torus using as a basis a periodic orbit and we extend this to a $2n-2$ dimensional object in the $2n-1$ dimensional energy surface. We present our methods using benchmark examples for two and three degree of freedom Hamiltonian systems to illustrate the corresponding algorithm for this construction. Towards this end we use the normal form quadratic Hamiltonian system with two and three degrees of freedom. We found that the periodic orbit dividing surface can provide us the same dynamical information as the dividing surface constructed using normally hyperbolic invariant manifolds. This is significant because, in general, computations of normally hyperbolic invariant manifolds are very difficult in Hamiltonian systems with three or more degrees of freedom. However, our method avoids this computation and the only information that we need is the location of one periodic orbit.
Comment: 26 pages, 8 figures, accepted for publication in the International Journal of Bifurcation and Chaos

الوصول الحر: http://arxiv.org/abs/2106.01141Test

المؤلفون: Katsanikas, M., Sanjuan, B. Aguilar, Montoya, F. Gonzalez, Garcia-Garrido, V., Wiggins, S.

مصطلحات موضوعية: Nonlinear Sciences - Chaotic Dynamics, Mathematics - Dynamical Systems, Physics - Chemical Physics

الوصف: In this paper, we explore the dynamics of a Hamiltonian system after a double van der Waals potential energy surface degenerates into a single well. The energy of the system is increased from the bottom of the potential well up to the dissociation energy, which occurs when the system becomes open. In particular, we study the bifurcations of the basic families of periodic orbits of this system as the energy increases using Lagrangian descriptors and Poincar\'e maps. We investigate the capability of Lagrangian descriptors to find periodic orbits of bifurcating families for the case of resonant, saddle-node and pitchfork bifurcations.
Comment: 17 pages, 14 figures

الوصول الحر: http://arxiv.org/abs/2105.07984Test

المؤلفون: Geng, Y., Katsanikas, M., Agaoglou, M., Wiggins, S.

مصطلحات موضوعية: Nonlinear Sciences - Chaotic Dynamics, Mathematics - Dynamical Systems, Physics - Chemical Physics, 34Cxx, 70Hxx

الوصف: In this work, we continue the study of the bifurcations of the critical points in a symmetric Caldera potential energy surface. In particular, we study the influence of the depth of the potential on the trajectory behavior before and after the bifurcations of the critical points. We observe two different types of trajectory behavior: dynamical matching and the non-existence of dynamical matching. Dynamical matching is a phenomenon that limits the way in which a trajectory can exit the Caldera based solely on how it enters the Caldera. Furthermore, we discuss two different types of symmetric Caldera potential energy surface and the transition from the one type to the other through the bifurcations of the critical points.

الوصول الحر: http://arxiv.org/abs/2105.05649Test

المؤلفون: García-Garrido, V. J., Wiggins, S.

مصطلحات موضوعية: Mathematics - Dynamical Systems, Nonlinear Sciences - Chaotic Dynamics, Physics - Chemical Physics, 70Kxx, 34Cxx, 70Hxx

الوصف: 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.
Comment: 14 pages, 9 figures

الوصول الحر: http://arxiv.org/abs/2105.00285Test

المؤلفون: Agaoglou, M., Garcia-Garrido, V. J., Katsanikas, M., Wiggins, S.

مصطلحات موضوعية: Nonlinear Sciences - Chaotic Dynamics, Mathematics - Dynamical Systems, Physics - Chemical Physics, 37N99, 70K44, 70H05, 70H07, 34C45, 34C37

الوصف: In this paper we demonstrate the capability of the method of Lagrangian descriptors to unveil the phase space structures that characterize transport in high-dimensional symplectic maps. In order to illustrate its use, we apply it to a four-dimensional symplectic map model that is used in chemistry to explore the nonlinear dynamics of van der Waals complexes. The advantage of this technique is that it allows us to easily and effectively extract the invariant manifolds that determine the dynamics of the system under study by means of examining the intersections of the underlying phase space structures with low-dimensional slices. With this approach, one can perform a full computational phase space tomography from which three-dimensional representations of the higher-dimensional phase space can be systematically reconstructed. This analysis may be of much help for the visualization and understanding of the nonlinear dynamical mechanisms that take place in high-dimensional systems. In this context, we demonstrate how this tool can be used to detect whether the stable and unstable manifolds of the system intersect forming turnstile lobes that enclose a certain phase space volume, and the nature of their intersection.
Comment: 24 pages

الوصول الحر: http://arxiv.org/abs/2103.06682Test

المؤلفون: Crossley, R., Agaoglou, M., Katsanikas, M., Wiggins, S.

مصطلحات موضوعية: Nonlinear Sciences - Chaotic Dynamics, Mathematics - Dynamical Systems, Physics - Chemical Physics, 37N99, 70K44, 70H05, 70H07, 34C45, 34C37

الوصف: In this paper we compare the method of Lagrangian descriptors with the classical method of Poincare maps for revealing the phase space structure of two degree-of-freedom Hamiltonian systems. The comparison is carried out by considering the dynamics of a two degree-of-freedom system having a valley ridge inflection point (VRI) potential energy surface. VRI potential energy surfaces have four critical points: a high energy saddle and a lower energy saddle separating two wells. In between the two saddle points is a valley ridge inflection point that is the point where the potential energy surface geometry changes from a valley to a ridge. The region between the two saddles forms a reaction channel and the dynamical issue of interest is how trajectories cross the high energy saddle, evolve towards the lower energy saddle, and select a particular well to enter. Lagrangian descriptors and Poincare maps are compared for their ability to determine the phase space structures that govern this dynamical process.
Comment: 19 pages

الوصول الحر: http://arxiv.org/abs/2102.04904Test

المؤلفون: Geng, Y., Katsanikas, M., Agaoglou, M., Wiggins, S.

مصطلحات موضوعية: Nonlinear Sciences - Chaotic Dynamics, Mathematics - Dynamical Systems, Physics - Chemical Physics, 34Cxx, 70Hxx

الوصف: Many organic chemical reactions are governed by potential energy surfaces that have a region with the topographical features of a caldera. If the caldera has a symmetry then trajectories transiting the caldera region are observed to exhibit a phenomenon that is referred to as dynamical matching. Dynamical matching is a constraint that restricts the way in which a trajectory can exit the caldera based solely on how it enters the caldera. In this paper we show that bifurcations of the critical points of the caldera potential energy surface can destroy dynamical matching even when the symmetry of the caldera is not affected by the bifurcation.
Comment: 11 pages, 8 figures

الوصول الحر: http://arxiv.org/abs/2102.02552Test

المؤلفون: García-Garrido, V. J., Katsanikas, M., Agaoglou, M., Wiggins, S.

مصطلحات موضوعية: Physics - Chemical Physics, Mathematics - Dynamical Systems, Nonlinear Sciences - Chaotic Dynamics, 34C37, 70K44, 34Cxx, 70Hxx

الوصف: 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.
Comment: 11 pages, 9 figures

الوصول الحر: http://arxiv.org/abs/2006.05969Test