Visualizing the perturbation of partial integrability

González, Federico; Jung, Christof

doi:10.1088/1751-8113/48/43/435101

Cited by 16 publications

(8 citation statements)

References 35 publications

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“…A few fragments around the tangentially stable and normally hyperbolic fixed points survive. This is similar to the scenario found in the example of Gonzalez & Jung (2015). For E = −2534 also the two orbits split off from the orbit Γv in its first pitchfork bifurcation become normally elliptic.…”

Section: Numerically Constructed Restricted Poincaré Map On the Nhimsupporting

confidence: 85%

“…the NHIM surface in the Poincaré map, acquires the topology of a sphere S 2 . This is very similar to the contraction of the boundary in the NHIM construction for the example of an electron in a perturbed magnetic dipole field as explained in Gonzalez & Jung (2015). To show plots of this restricted map we have to project the NHIM surface in some form.…”

Section: Numerically Constructed Restricted Poincaré Map On the Nhimmentioning

confidence: 53%

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Orbital and escape dynamics in barred galaxies – II. The 3D system: exploring the role of the normally hyperbolic invariant manifolds

Jung¹,

Zotos²

2016

Mon. Not. R. Astron. Soc.

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A three degrees of freedom (3-dof) barred galaxy model composed of a spherically symmetric nucleus, a bar, a flat disc and a spherically symmetric dark matter halo is used for investigating the dynamics of the system. We use colour-coded plots to demonstrate how the value of the semi-major axis of the bar influences the regular or chaotic dynamics of the 3-dof system. For distinguishing between ordered and chaotic motion we use the Smaller ALingment Index (SALI) method, a fast yet very accurate tool. Undoubtedly, the most important elements of the dynamics are the normally hyperbolic invariant manifolds (NHIMs) located in the vicinity of the index 1 Lagrange points L 2 and L 3 . These manifolds direct the flow of stars over the saddle points, while they also trigger the formation of rings and spirals. The dynamics in the neighbourhood of the saddle points is visualized by bifurcation diagrams of the Lyapunov orbits as well as by the restriction of the Poincaré map to the NHIMs. In addition, we reveal how the semi-major axis of the bar influences the structure of these manifolds which determine the final stellar structure (rings or spirals). Our numerical simulations suggest that in galaxies with weak bars the formation of R 1 rings or R 1 pseudo-rings is favoured. In the case of galaxies with intermediate and strong bars the invariant manifolds seem to give rise to R 1 R 2 rings and twin spiral formations, respectively. We also compare our numerical outcomes with earlier related work and with observational data.

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Section: Numerically Constructed Restricted Poincaré Map On the Nhimsupporting

confidence: 85%

Section: Numerically Constructed Restricted Poincaré Map On the Nhimmentioning

confidence: 53%

Orbital and escape dynamics in barred galaxies – II. The 3D system: exploring the role of the normally hyperbolic invariant manifolds

Jung¹,

Zotos²

2016

Mon. Not. R. Astron. Soc.

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“…Therefore the restriction of the Poincaré map to this NHIM surface makes sense and it is a 2-dimensional Poincaré map which can be represented by 2-dimensional graphics. For previous examples of the construction and use of this restricted map see Gonzalez et al (2014); Gonzalez & Jung (2015) and Paper III. Since NHIMs are persistent under perturbations also the existence of this map is persistent under perturbations and is an interesting tool to study the perturbation scenario in 3-dof systems by 2-dimensional graphics.…”

Section: The Numerical Restricted Map On the Nhimmentioning

confidence: 99%

“…Therefore, also the development scenario of the Lyapunov orbits will be investigated in detail. We apply recently developed methods (e.g., Gonzalez et al 2014;Gonzalez & Jung 2015) for the numerical study of NHIMs and of the Poincaré map restricted to the NHIM. This restricted map acts on a 2-dimensional domain, can be represented by 2-dimensional graphics and therefore it is an ideal tool to present graphically the development scenario of codimension 2 NHIMs in 3-dof systems.…”

Section: Introductionmentioning

confidence: 99%

Unravelling the escape dynamics and the nature of the normally hyperbolic invariant manifolds in tidally limited star clusters

Zotos¹,

Jung²

2016

Mon. Not. R. Astron. Soc.

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The escape mechanism of orbits in a star cluster rotating around its parent galaxy in a circular orbit is investigated. A three degrees of freedom model is used for describing the dynamical properties of the Hamiltonian system. The gravitational field of the star cluster is represented by a smooth and spherically symmetric Plummer potential. We distinguish between ordered and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. The Smaller Alignment Index (SALI) method is used for determining the regular or chaotic nature of the orbits. The basins of escape are located and they are also correlated with the corresponding escape time of the orbits. Areas of bounded regular or chaotic motion and basins of escape were found to coexist in the (x, z) plane. The properties of the normally hyperbolic invariant manifolds (NHIMs), located in the vicinity of the index-1 Lagrange points L 1 and L 2 , are also explored. These manifolds are of paramount importance as they control the flow of stars over the saddle points, while they also trigger the formation of tidal tails observed in star clusters. Bifurcation diagrams of the Lyapunov periodic orbits as well as restrictions of the Poincaré map to the NHIMs are deployed for elucidating the dynamics in the neighbourhood of the saddle points. The extended tidal tails, or tidal arms, formed by stars with low velocity which escape through the Lagrange points are monitored. The numerical results of this work are also compared with previous related work. ration process because of the movement of stars above the escape velocity and the time-scale of relaxation which determines the rate of the dynamical evolution of a star cluster were the first which have been investigated analytically (e.g., Ambartsumian 1938; Spitzer 1940). Later on, the dynamical properties of stars which have been scattered above the critical escape energy were examined (King 1959), while a year later the importance of the close encounters between stars for the rate of mass loss of a star cluster was studied (Hénon 1960).According to Lada & Lada (2003) a considerable amount of the stars of a galaxy are actually born in star clusters. In particular, some recent studies shed some light on the mechanism of how stars are in fact born and develop star clusters (e.g., Bressert et al. 2010;Kruijssen 2012). Inevitably, all star clusters dissolve over time due to two main reasons: (i) the two-body process (encounters) that affects all the members of a cluster thus forcing them to obtain escape velocities and (ii) the strong tidal forces due to the

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“…This is possible along the following ideas: The relevant NHIMs in the full dimensional Poincaré map are invariant subsets of dimension 2 and the restriction of the Poincaré map to a NHIM exists and has as domain the 2 dimensional NHIM surface itself. So we can study this restricted map and it is the ideal tool to present the saddle dynamics and its development scenario by 2-dimensional graphics (e.g., [9,10]). It includes a graphical representation of the development scenario of the Lyapunov orbits and of some further important periodic orbits, related to the Lyapunov orbits.…”

Section: Introductionmentioning

confidence: 99%

Correlating the escape dynamics and the role of the normally hyperbolic invariant manifolds in a binary system of dwarf spheroidal galaxies

Zotos

Jung

2018

International Journal of Non-Linear Mechanics

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We elucidate the escape properties of stars moving in the combined gravitational field of a binary system of two interacting dwarf spheroidal galaxies. A galaxy model of three degrees of freedom is adopted for describing the dynamical properties of the Hamiltonian system. All the numerical values of the involved parameters are chosen having in mind the real binary system of the dwarf spheroidal galaxies NGC 147 and NGC 185. We distinguish between bounded (regular, sticky or chaotic) and escaping motion by classifying initial conditions of orbits in several types of two dimensional planes, considering only unbounded motion for several energy levels. We analyze the orbital structure of all types of two dimensional planes of initial conditions by locating the basins of escape and also by measuring the corresponding escape time of the orbits. Furthermore, the properties of the normally hyperbolic invariant manifolds (NHIMs), located in the vicinity of the index-1 saddle points L 1 , L 2 , and L 3 , are also investigated. These manifolds are of great importance, as they control the flow of stars (between the two galaxies and toward the exterior region) over the different saddle points. In addition, bifurcation diagrams of the Lyapunov periodic orbits as well as restrictions of the Poincaré map to the NHIMs are presented for revealing the dynamics in the neighbourhood of the saddle points. Comparison between the current outcomes and previous related results is also made.

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Visualizing the perturbation of partial integrability

Cited by 16 publications

References 35 publications

Orbital and escape dynamics in barred galaxies – II. The 3D system: exploring the role of the normally hyperbolic invariant manifolds

Orbital and escape dynamics in barred galaxies – II. The 3D system: exploring the role of the normally hyperbolic invariant manifolds

Unravelling the escape dynamics and the nature of the normally hyperbolic invariant manifolds in tidally limited star clusters

Correlating the escape dynamics and the role of the normally hyperbolic invariant manifolds in a binary system of dwarf spheroidal galaxies

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