Sampling Phase Space Dividing Surfaces Constructed from Normally Hyperbolic Invariant Manifolds (NHIMs)

2018 ◽  
Vol 122 (42) ◽  
pp. 8354-8362 ◽  
Author(s):  
Gregory S. Ezra ◽  
Stephen Wiggins
Keyword(s):  
Phase Space ◽  
Invariant Manifolds ◽  
Normally Hyperbolic Invariant Manifolds

Invariant Manifold Detection From Phase Space Trajectories

2008 ◽  
Author(s):  
Joseph Kuehl ◽  
David Chelidze
Keyword(s):  
Phase Space ◽  
Invariant Manifold ◽  
Invariant Manifolds ◽  
Basins Of Attraction ◽  
Detection Methods ◽  
Trajectory Data ◽  
Space Trajectories ◽  
Data Set ◽  
Phase Space Warping ◽  
Phase Space Trajectories

Invariant manifolds provide important information about the structure of flows. When basins of attraction are present, the stable invariant manifold serves as the boundary between these basins. Thus, in experimental applications such as vibrations problems, knowledge of these manifolds is essential to understanding the evolution of phase space trajectories. Most existing methods for identifying invariant manifolds of a flow rely on knowledge of the flow field. However, in experimental applications only knowledge of phase space trajectories is available. We provide modifications to several existing invariant manifold detection methods which enables them to deal with trajectory only data, as well as introduce a new method based on the concept of phase space warping. The method of Stochastic Interrogation applied to the damped, driven Duffing equation is used to generate our data set. The result is a set of trajectory data which randomly populates a phase space. Manifolds are detected from this data set using several different methods. First is a variation on manifold “growing,” and is based on distance of closest approach to a hyperbolic trajectory with “saddle like behavior.” Second, three stretching based schemes are considered. One considers the divergence of trajectory pairs, another quantifies the deformation of a nearest neighbor cloud, and the last uses flow fields calculated from the trajectory data. Finally, the new phase space warping method is introduced. This method takes advantage of the shifting (warping) experienced by a phase space as the parameters of the system are slightly varied. This results in a shift of the invariant manifolds. The region spanned by this shift, provides a means to identify the invariant manifolds. Results show that this method gives superior detection and is robust with respect to the amount of data.

scholarly journals Tilting and Squeezing: Phase Space Geometry of Hamiltonian Saddle-Node Bifurcation and its Influence on Chemical Reaction Dynamics

2020 ◽  
Vol 30 (04) ◽  
pp. 2030008 ◽  
Author(s):  
Víctor J. García-Garrido ◽  
Shibabrat Naik ◽  
Stephen Wiggins
Keyword(s):  
Phase Space ◽  
Hamiltonian System ◽  
Potential Energy ◽  
Invariant Manifolds ◽  
Reaction Dynamics ◽  
Potential Energy Function ◽  
Space Structures ◽  
Stable And Unstable Manifolds ◽  
Saddle Node Bifurcation ◽  
Saddle Node

In this article, we present the influence of a Hamiltonian saddle-node bifurcation on the high-dimensional phase space structures that mediate reaction dynamics. To achieve this goal, we identify the phase space invariant manifolds using Lagrangian descriptors, which is a trajectory-based diagnostic suitable for the construction of a complete “phase space tomography” by means of analyzing dynamics on low-dimensional slices. First, we build a Hamiltonian system with one degree-of-freedom (DoF) that models reaction, and study the effect of adding a parameter to the potential energy function that controls the depth of the well. Then, we extend this framework to a saddle-node bifurcation for a two DoF Hamiltonian, constructed by coupling a harmonic oscillator, i.e. a bath mode, to the other reactive DoF in the system. For this problem, we describe the phase space structures associated with the rank-1 saddle equilibrium point in the bottleneck region, which is a Normally Hyperbolic Invariant Manifold (NHIM) and its stable and unstable manifolds. Finally, we address the qualitative changes in the reaction dynamics of the Hamiltonian system due to changes in the well depth of the potential energy surface that gives rise to the saddle-node bifurcation.

Catastrophes with indeterminate outcome

1991 ◽  
Vol 432 (1884) ◽  
pp. 113-123 ◽  
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Invariant Manifolds ◽  
Space Geometry ◽  
Dissipative Dynamical System ◽  
Forced Oscillators ◽  
Phase Space Geometry

A catastrophe in a dissipative dynamical system which causes an attractor to completely lose stability will result in a transient trajectory making a rapid jump in phase space to some other attractor. In systems where more than one other attractor is available, the attractor chosen may depend very sensitively on how the catastrophe is realized. Two examples in forced oscillators of Duffing type illustrate how the probabilities of different outcomes can be estimated using the phase space geometry of invariant manifolds.

scholarly journals Phase Space Structure and Transport in a Caldera Potential Energy Surface

2018 ◽  
Vol 28 (13) ◽  
pp. 1830042 ◽  
Author(s):  
Matthaios Katsanikas ◽  
Stephen Wiggins
Keyword(s):  
Phase Space ◽  
Potential Energy ◽  
Potential Energy Surface ◽  
Periodic Orbits ◽  
Central Region ◽  
Invariant Manifolds ◽  
Energy Surface ◽  
Unstable Periodic Orbits ◽  
Phase Space Transport ◽  
The Family

We study phase space transport in a 2D caldera potential energy surface (PES) using techniques from nonlinear dynamics. The caldera PES is characterized by a flat region or shallow minimum at its center surrounded by potential walls and multiple symmetry related index one saddle points that allow entrance and exit from this intermediate region. We have discovered four qualitatively distinct cases of the structure of the phase space that govern phase space transport. These cases are categorized according to the total energy and the stability of the periodic orbits associated with the family of the central minimum, the bifurcations of the same family, and the energetic accessibility of the index one saddles. In each case, we have computed the invariant manifolds of the unstable periodic orbits of the central region of the potential, and the invariant manifolds of the unstable periodic orbits of the families of periodic orbits associated with the index one saddles. The periodic orbits of the central region are, for the first case, the unstable periodic orbits with period 10 that are outside the stable region of the stable periodic orbits of the family of the central minimum. In addition, the periodic orbits of the central region are, for the second and third cases, the unstable periodic orbits of the family of the central minimum and for the fourth case the unstable periodic orbits with period 2 of a period-doubling bifurcation of the family of the central minimum. We have found that there are three distinct mechanisms determined by the invariant manifold structure of the unstable periodic orbits that govern the phase space transport. The first mechanism explains the nature of the entrance of the trajectories from the region of the low energy saddles into the caldera and how they may become trapped in the central region of the potential. The second mechanism describes the trapping of the trajectories that begin from the central region of the caldera, their transport to the regions of the saddles, and the nature of their exit from the caldera. The third mechanism describes the phase space geometry responsible for the dynamical matching of trajectories originally proposed by Carpenter and described in [Collins et al., 2014] for the two-dimensional caldera PES that we consider.

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