Experimental validation of phase space conduits of transition between potential wells

Ross, Shane D.; BozorgMagham, Amir E.; Naik, Shibabrat; Virgin, Lawrence N.

doi:10.1103/physreve.98.052214

Cited by 24 publications

(14 citation statements)

References 38 publications

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“…The mathematical model of a rolling ball on a stationary surface was established in [33]. Experiments [26,34] regarding escape from the potential wells on similar surfaces were shown to validate the theory of the phase space conduits predicted by the mathematical model, which mediate the transitions between wells in the system. The dissipation of energy cannot be avoided in any physical experiment, but over small enough time-scales of interest, [26] justified that dissipation could be ignored.…”

Section: Ball Rolling On a Stationary Surfacementioning

confidence: 85%

“…Some researchers have studied escape in conservative gyroscopic systems (e.g., [6,25]). There exist transition tubes controlling the escape which are topologically the same as in an inertial system [2,8,26]. However, to the best knowledge of the authors, no study has been carried out to study the escape in systems with both dissipative and gyroscopic forces present.…”

Section: Introductionmentioning

confidence: 99%

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Geometry of escape and transition dynamics in the presence of dissipative and gyroscopic forces in two degree of freedom systems

Zhong

Ross

2020

Communications in Nonlinear Science and Numerical Simulation

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Escape from a potential well can occur in different physical systems, such as capsize of ships, resonance transitions in celestial mechanics, and dynamic snap-through of arches and shells, as well as molecular reconfigurations in chemical reactions. The criteria and routes of escape in one-degree of freedom systems has been well studied theoretically with reasonable agreement with experiment. The trajectory can only transit from the hilltop of the one-dimensional potential energy surface. The situation becomes more complicated when the system has higher degrees of freedom since it has multiple routes to escape through an equilibrium of saddle-type, specifically, an index-1 saddle. This paper summarizes the geometry of escape across a saddle in some widely known physical systems with two degrees of freedom and establishes the criteria of escape providing both a methodology and results under the conceptual framework known as tube dynamics. These problems are classified into two categories based on whether the saddle projection and focus projection in the symplectic eigenspace are coupled or not when damping and/or gyroscopic effects are considered. For simplicity, only the linearized system around the saddle points is analyzed, but the results generalize to the nonlinear system. We define a transition region, T h , as the region of initial conditions of a given initial energy h which transit from one side of a saddle to the other. We find that in conservative systems, the boundary of the transition region, ∂T h , is a cylinder, while in dissipative systems, ∂T h is an ellipsoid.

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Section: Ball Rolling On a Stationary Surfacementioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Geometry of escape and transition dynamics in the presence of dissipative and gyroscopic forces in two degree of freedom systems

Zhong

Ross

2020

Communications in Nonlinear Science and Numerical Simulation

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“…A detailed analysis of the changes in the stability and related manifestation of the reaction mechanism is beyond the scope of this article and will be the focus of future work. The methods used for computing the phase space structures are described in the supplemental material (see [ 60 ] for more details on a similar PES). We show the homoclinic tangles for sample values of the total energy and coupling strength.…”

Section: Classical Dynamics: Dynamical Murrell – Laidler Mechanismandmentioning

confidence: 99%

Classical and Quantum Dynamical Manifestations of Index-$$2$$ Saddles: Concerted Versus Sequential Reaction Mechanisms

2021

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The presence of higher-index saddles on a multidimensional potential energy surface is usually assumed to be of little significance in chemical reaction dynamics. Such a viewpoint requires careful reconsideration, thanks to elegant experiments and novel theoretical approaches that have come about in recent years. In this work, we perform a detailed classical and quantum dynamical study of a model two-degree-of-freedom Hamiltonian, which captures the essence of the debate regarding the dominance of a concerted or a stepwise reaction mechanism. We show that the ultrafast shift of the mechanism from a concerted to a stepwise one is essentially a classical dynamical effect. In addition, due to the classical phase space being a mixture of regular and chaotic dynamics, it is possible to have a rich variety of dynamical behavior, including a Murrell – Laidler type mechanism, even at energies sufficiently above that of the index- saddle. We rationalize the dynamical results using an explicit construction of the classical invariant manifolds in the phase space. Supplementary Information The online version contains supplementary material with details on computation of the invariant manifolds which available at 10.1134/S1560354721020052.

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“…The deterministic evolution of a system between two stable states through an intermediate unstable state is a fundamental setting for an important form of dynamical evolution that informs our way of thinking of "transition phenomena". This dynamical situation is common to many diverse fields in science and engineering, such as celestial mechanics [1,2], structural mechanics [3][4][5], and chemical reaction dynamics. This setting is natural for studies in the latter area and will be the focus of the applications that we consider.…”

Section: Introductionmentioning

confidence: 99%

Reactive Islands for Three Degrees-of-Freedom Hamiltonian Systems

Krajňák,

García-Garrido,

Wiggins

2021

Preprint

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We develop the geometrical, analytical, and computational framework for reactive island theory for three degrees-of-freedom time-independent Hamiltonian systems. In this setting, the dynamics occurs in a 5-dimensional energy surface in phase space and is governed by four-dimensional stable and unstable manifolds of a three-dimensional normally hyperbolic invariant sphere. The stable and unstable manifolds have the geometrical structure of spherinders and we provide the means to investigate the ways in which these spherinders and their intersections determine the dynamical evolution of trajectories. This geometrical picture is realized through the computational technique of Lagrangian descriptors. In a set of trajectories, Lagrangian descriptors allow us to identify the ones closest to a stable or unstable manifold. Using an approximation of the manifold on a surface of section we are able to calculate the flux between two regions of the energy surface.

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Experimental validation of phase space conduits of transition between potential wells

Cited by 24 publications

References 38 publications

Geometry of escape and transition dynamics in the presence of dissipative and gyroscopic forces in two degree of freedom systems

Geometry of escape and transition dynamics in the presence of dissipative and gyroscopic forces in two degree of freedom systems

Classical and Quantum Dynamical Manifestations of Index-$$2$$ Saddles: Concerted Versus Sequential Reaction Mechanisms

Reactive Islands for Three Degrees-of-Freedom Hamiltonian Systems

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