Chaos and Noise in Galactic Potentials

Habib, Salman; Kandrup, Henry E.; Mahon, M. Elaine

doi:10.1086/303935

Cited by 43 publications

(60 citation statements)

References 55 publications

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“…2) a conclusion that can be motivated theoretically [29] and and has been confirmed computationally [13]. It is interesting that this two-stage evolution -an epoch without escapes followed by an epoch with escapes apparently sampling a Poisson process -can also be observed in the absence of noise if one considers a strongly localised ensemble of initial conditions trapped near a regular island and ascertains the time at which each member of the ensemble escapes.…”

Section: White Noisementioning

confidence: 68%

Noise-induced phase space transport in two-dimensional Hamiltonian systems

Pogorelov

Kandrup

1999

Phys. Rev. E

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First passage time experiments were used to explore the effects of low amplitude noise as a source of accelerated phase space diffusion in two-dimensional Hamiltonian systems, and these effects were then compared with the effects of periodic driving. The objective was to quantify and understand the manner in which "sticky" chaotic orbits that, in the absence of perturbations, are confined near regular islands for very long times, can become "unstuck" much more quickly when subjected to even very weak perturbations. For both noise and periodic driving, the typical escape time scales logarithmically with the amplitude of the perturbation. For white noise, the details seem unimportant: Additive and multiplicative noise typically have very similar effects, and the presence or absence of a friction related to the noise by a Fluctuation-Dissipation Theorem is also largely irrelevant. Allowing for colored noise can significantly decrease the efficacy of the perturbation, but only when the autocorrelation time, which vanishes for white noise, becomes so large that there is little power at frequencies comparable to the natural frequencies of the unperturbed orbit. Similarly, periodic driving is relatively inefficient when the driving frequency is not comparable to these natural frequencies. This suggests strongly that noise-induced extrinsic diffusion, like modulational diffusion associated with periodic driving, is a resonance phenomenon. The logarithmic dependence of the escape time on amplitude reflects the fact that the time required for perturbed and unperturbed orbits to diverge a given distance scales logarithmically in the amplitude of the perturbation. PACS number(s): 05.60.+w, 51.10.+y, 05.40.+j I. MOTIVATIONIt is well known that a complex phase space containing large measures of both regular and chaotic orbits is often partitioned by such partial obstructions as cantori [1] or Arnold webs [2] which, although not serving as absolute barriers, can significantly impede the motion of a chaotic orbit through a connected phase space region. Indeed, the fact that, in two-dimensional Hamiltonian systems, chaotic orbits can be "stuck" near regular islands for very long times was discovered empirically [3] long before the existence of cantori was proven [4].It has also been long known that low amplitude stochastic perturbations can accelerate Hamiltonian phase space transport by enabling orbits to traverse these partial barriers. This was, e.g., explored by Lieberman and Lichtenberg [5], who investigated how motion described by the simplified Ulam version of the Fermi acceleration map [6] is impacted by random perturbations, allowing for the modified equations [7] where the "noise" ∆ψ corresponds to a random phase shift uniformly sampling an interval [−ϕ, +ϕ].That stochastic perturbations can have such effects on Hamiltonian systems is important in understanding the limitations of simple models of real systems. In the absence of all "perturbations" and any other irregularities, the chaotic phase space associated with so...

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Section: White Noisementioning

confidence: 68%

Noise-induced phase space transport in two-dimensional Hamiltonian systems

Pogorelov

Kandrup

1999

Phys. Rev. E

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“…In the last few years, the study of individual orbits in some galactic potential is a new branch of galactic dynamics (see, for example Ref. [10][11][12][13].…”

Section: Introduction and Statements Of Main Resultsmentioning

confidence: 99%

Periodic orbits of the generalized Friedmann-Robertson-Walker potential in galactic dynamics in a rotating reference frame

2017

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“…The two-dimensional limit of equation (3) appropriate for motion with z ; 0 served as a prototypical example in several earlier papers (Mahon et al 1995;Habib et al 1997), which discuss its physical properties extensively. The fully three-dimensional version was explored in the context of chaotic mixing in Kandrup (1998).…”

Section: P H a S E -S Pac E T R A N S P O R T I N C H Ao T I C H A M mentioning

confidence: 99%

Chaotic mixing in noisy Hamiltonian systems

Kandrup

Pogorelov

Sideris

2000

Monthly Notices of the Royal Astronomical Society

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This paper summarizes an investigation of the effects of low-amplitude noise and periodic driving on phase-space transport in three-dimensional Hamiltonian systems, a problem directly applicable to systems like galaxies, where such perturbations reflect internal irregularities and/or a surrounding environment. A new diagnostic tool is exploited to quantify the extent to which, over long times, different segments of the same chaotic orbit evolved in the absence of such perturbations can exhibit very different amounts of chaos. First-passage-time experiments are used to study how small perturbations of an individual orbit can dramatically accelerate phase-space transport, allowing`sticky' chaotic orbits trapped near regular islands to become unstuck on surprisingly short time-scales. The effects of small perturbations are also studied in the context of orbit ensembles with the aim of understanding how such irregularities can increase the efficacy of chaotic mixing. For both noise and periodic driving, the effect of the perturbation scales roughly logarithmically in amplitude. For white noise, the details are unimportant: additive and multiplicative noise tend to have similar effects and the presence or absence of friction related to the noise by a fluctuation±dissipation theorem is largely irrelevant. Allowing for coloured noise can significantly decrease the efficacy of the perturbation, but only when the autocorrelation time, which vanishes for white noise, becomes so large that there is little power at frequencies comparable with the natural frequencies of the unperturbed orbit. This suggests strongly that noise-induced extrinsic diffusion, like modulational diffusion associated with periodic driving, is a resonance phenomenon. Potential implications for galaxies are discussed.

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Chaos and Noise in Galactic Potentials

Cited by 43 publications

References 55 publications

Noise-induced phase space transport in two-dimensional Hamiltonian systems

Noise-induced phase space transport in two-dimensional Hamiltonian systems

Periodic orbits of the generalized Friedmann-Robertson-Walker potential in galactic dynamics in a rotating reference frame

Chaotic mixing in noisy Hamiltonian systems

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