scholarly journals Detection of Dynamical Matching in a Caldera Hamiltonian System Using Lagrangian Descriptors

2020 ◽  
Vol 30 (09) ◽  
pp. 2030026
Author(s):  
M. Katsanikas ◽  
Víctor J. García-Garrido ◽  
S. Wiggins
Keyword(s):  
Phase Space ◽  
Energy Surface ◽  
Critical Value ◽  
Unstable Periodic Orbits ◽  
The Family ◽  
Heteroclinic Connection ◽  
Use Phase ◽  
Unstable Manifolds ◽  
Type Potential ◽  
Lagrangian Descriptors

The goal of this paper is to apply the method of Lagrangian descriptors to reveal the phase space mechanism by which a Caldera-type potential energy surface (PES) exhibits the dynamical matching phenomenon. Using this technique, we can easily establish that the nonexistence of dynamical matching is a consequence of heteroclinic connections between the unstable manifolds of the unstable periodic orbits (UPOs) of the upper index-1 saddles (entrance channels to the Caldera) and the stable manifolds of the family of UPOs of the central minimum of the Caldera, resulting in the temporary trapping of trajectories. Moreover, dynamical matching will occur when there is no heteroclinic connection, which allows trajectories to enter and exit the Caldera without interacting with the shallow region of the central minimum. Knowledge of this phase space mechanism is relevant because it allows us to effectively predict the existence, and nonexistence, of dynamical matching. In this work, we explore a stretched Caldera potential by means of Lagrangian descriptors, allowing us to accurately compute the critical value for the stretching parameter for which dynamical matching behavior occurs in the system. This approach is shown to provide a tremendous advantage for exploring this mechanism in comparison to other methods from nonlinear dynamics that use phase space dividing surfaces.

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.

Deconstructing field-induced ketene isomerization through Lagrangian descriptors

2016 ◽  
Vol 18 (5) ◽  
pp. 4008-4018 ◽  
Author(s):  
Galen T. Craven ◽  
Rigoberto Hernandez
Keyword(s):  
Phase Space ◽  
Potential Energy ◽  
Potential Energy Surface ◽  
Energy Surface ◽  
Reaction Path ◽  
State Transitions ◽  
Lagrangian Descriptors

Phase space contours (shown in color) constructed using the method of Lagrangian descriptors resolve the separatrices governing state transitions on the reaction-path potential energy surface (shown in white) for field-induced ketene isomerization.

scholarly journals Phase Space Analysis of the Nonexistence of Dynamical Matching in a Stretched Caldera Potential Energy Surface

2019 ◽  
Vol 29 (04) ◽  
pp. 1950057 ◽  
Author(s):  
Matthaios Katsanikas ◽  
Stephen Wiggins
Keyword(s):  
Phase Space ◽  
Potential Energy ◽  
Potential Energy Surface ◽  
Energy Surface ◽  
Critical Value ◽  
Space Structure ◽  
Space Structures ◽  
Two Dimensional ◽  
Phase Space Structure ◽  
Parametrized Family

In this paper, we continue our studies of the two-dimensional caldera potential energy surface in a parametrized family that allows for a study of the effect of symmetry on the phase space structures that govern how trajectories enter, cross, and exit the region of the caldera. As a particular form of trajectory crossing, we are able to determine the effect of symmetry and phase space structure on dynamical matching. We show that there is a critical value of the symmetry parameter which controls the phase space structures responsible for the manner of crossing, interacting with the central region (including trapping in this region) and exiting the caldera. We provide an explanation for the existence of this critical value in terms of the behavior of the Hénon stability parameter for the associated periodic orbits.

Detection of Phase Space Structures of the Cat Map with Lagrangian Descriptors

2018 ◽  
Vol 23 (6) ◽  
pp. 751-766 ◽  
Author(s):  
Víctor J. García-Garrido ◽  
Francisco Balibrea-Iniesta ◽  
Stephen Wiggins ◽  
Ana M. Mancho ◽  
Carlos Lopesino
Keyword(s):  
Phase Space ◽  
Space Structures ◽  
Lagrangian Descriptors

scholarly journals Conditional Entropy: A Tool to Explore the Phase Space

1999 ◽  
Vol 172 ◽  
pp. 195-209
Author(s):  
P. Cincotta ◽  
C. Simó
Keyword(s):  
Phase Space ◽  
Chaotic Motion ◽  
Conditional Entropy ◽  
Characteristic Number ◽  
Random Variable ◽  
Arc Length ◽  
Unstable Periodic Orbits ◽  
Long Time ◽  
Stable Periodic ◽  
Lyapunov Characteristic Number

AbstractIn this paper we show that the Conditional Entropy of nearby orbits may be a useful tool to explore the phase space associated to a given Hamiltonian. The arc length parameter along the orbits, instead of the time, is used as a random variable to compute the entropy. In the first part of this work we summarise the main analytical results to support this tool while, in the second part, we present numerical evidence that this technique is able to localise (stable) periodic and quasiperiodic orbits, ‘aperiodic’ orbits (chaotic motion) and unstable periodic orbits (the ‘source’ of chaotic motion). Besides, we show that this technique provides a measure of chaos which is similar to that given by the largest Lyapunov Characteristic Number. It is important to remark that this method is very simple to compute and does not require long time integrations, just realistic physical times.

Hausdorff Dimension of Sets of Escaping Points and Escaping Parameters for Elliptic Functions

2015 ◽  
Vol 59 (3) ◽  
pp. 671-690
Author(s):  
Piotr Gałązka ◽  
Janina Kotus
Keyword(s):  
Hausdorff Dimension ◽  
Elliptic Function ◽  
Meromorphic Functions ◽  
Elliptic Functions ◽  
Critical Value ◽  
Parameter Plane ◽  
Dynamical Plane ◽  
The Family ◽  
Maximal Multiplicity ◽  
Additional Assumptions

AbstractLetbe a non-constant elliptic function. We prove that the Hausdorff dimension of the escaping set offequals 2q/(q+1), whereqis the maximal multiplicity of poles off. We also consider theescaping parametersin the familyfβ=βf, i.e. the parametersβfor which the orbit of one critical value offβescapes to infinity. Under additional assumptions onfwe prove that the Hausdorff dimension of the set of escaping parametersεin the familyfβis greater than or equal to the Hausdorff dimension of the escaping set in the dynamical space. This demonstrates an analogy between the dynamical plane and the parameter plane in the class of transcendental meromorphic functions.

scholarly journals Bifurcation of Dividing Surfaces Constructed from a Pitchfork Bifurcation of Periodic Orbits in a Symmetric Potential Energy Surface with a Post-Transition-State Bifurcation

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
M. Katsanikas ◽  
M. Agaoglou ◽  
S. Wiggins
Keyword(s):  
Potential Energy ◽  
Potential Energy Surface ◽  
Periodic Orbits ◽  
Lower Energy ◽  
Degrees Of Freedom ◽  
Energy Surface ◽  
Pitchfork Bifurcation ◽  
High Energy ◽  
Symmetric Potential ◽  
The Family

In this work, we analyze the bifurcation of dividing surfaces that occurs as a result of a pitchfork bifurcation of periodic orbits in a two degrees of freedom Hamiltonian System. The potential energy surface of the system that we consider has four critical points: two minima, a high energy saddle and a lower energy saddle separating two wells (minima). In this paper, we study the structure, the range, and the minimum and maximum extent of the periodic orbit dividing surfaces of the family of periodic orbits of the lower saddle as a function of the total energy.

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