95 research outputs found

### Orbital structure in oscillating galactic potentials

This paper focuses on symmetric potentials subjected to periodic driving.
Four unperturbed potentials V_0(r) were considered, namely the Plummer
potential and Dehnen potentials with \gamma=0.0, 0.5, and 1.0, each subjected
to a time-dependence of the form V(r,t)=V_0(r)(1+m_0\sin(\omega t)). In each
case, the orbits divide clearly into regular and chaotic, distinctions which
appear absolute. In particular, transitions from regularity to chaos are
seemingly impossible. Over finite time intervals, chaotic orbits subdivide into
what can be termed `sticky' chaotic orbits, which exhibit no large scale
secular changes in energy and remain trapped in the phase space region where
they started; and `wildly chaotic' orbits, which do exhibit systematic drifts
in energy as the orbits diffuse to inhabit different phase space regions. This
latter distinction is not absolute, apparently corresponding instead to orbits
penetrating a `leaky' phase space barrier. The three different orbit types can
be identified simply in terms of the frequencies for which their Fourier
spectra have the most power. An examination of the statistical properties of
orbit ensembles as a function of driving frequency \omega allows one to
identify the specific resonances that determine orbital structure. Attention
focuses also on how, for fixed amplitude m_0, such quantities as the mean
energy shift, the relative measure of chaotic orbits, and the mean value of the
largest Lyapunov exponent vary with driving frequency \omega and how, for fixed
\omega, the same quantities depend on m_0.Comment: 16 pages, 9 figures. Accepted for publication to MNRAS. Minor
editions and deletions. Updated reference

### Energy relaxation in galaxies induced by an external environment and/or incoherent internal pulsations

This paper explores the phenomenon of energy relaxation for stars in a galaxy
embedded in a high density environment that is subjected continually to
perturbations reflecting the presence of other nearby galaxies and/or
incoherent internal pulsations. The analysis is similar to earlier analyses of
energy relaxation induced by binary encounters between nearby stars and between
stars and giant molecular clouds in that the perturbations are idealised as a
sum of near-random events which can be modeled as diffusion and dynamical
friction. However, the analysis differs in one important respect: because the
time scale associated with these perturbations need not be short compared with
the characteristic dynamical time t_D for stars in the original galaxy, the
diffusion process cannot be modeled as resulting from a sequence of
instantaneous kicks, i.e., white noise. Instead, the diffusion is modeled as
resulting from random kicks of finite duration, i.e., coloured noise
characterised by a nonzero autocorrelation time t_c. A detailed analysis of
coloured noise generated by sampling an Ornstein-Uhlenbeck process leads to a
simpling scaling in terms of t_c and an effective diffusion constant D.
Interpreting D and t_c following early work by Chandrasekhar (1941) (the
`nearest neighbour approximation') implies that, for realistic choices of
parameter values, energy relaxation associated with an external environment
and/or internal pulsations could be important on times short compared with the
age of the Universe.Comment: 7 pages including 4 Figure

### Smooth potential chaos and N-body simulations

Integrations in fixed N-body realisations of smooth density distributions
corresponding to a chaotic galactic potential can be used to derive reliable
estimates of the largest (finite time) Lyapunov exponent X_S associated with an
orbit in the smooth potential generated from the same initial condition, even
though the N-body orbit is typically characterised by an N-body exponent X_N >>
X_S. This can be accomplished either by comparing initially nearby orbits in a
single N-body system or by tracking orbits with the same initial condition
evolved in two different N-body realisations of the same smooth density.Comment: 9 pages plus 7 figures, expanded version to appear in Astrophysical
Journa

### Chaos and the continuum limit in nonneutral plasmas and charged particle beams

This paper examines discreteness effects in nearly collisionless N-body
systems of charged particles interacting via an unscreened r^-2 force, allowing
for bulk potentials admitting both regular and chaotic orbits. Both for
ensembles and individual orbits, as N increases there is a smooth convergence
towards a continuum limit. Discreteness effects are well modeled by Gaussian
white noise with relaxation time t_R = const * (N/log L)t_D, with L the Coulomb
logarithm and t_D the dynamical time scale. Discreteness effects accelerate
emittance growth for initially localised clumps. However, even allowing for
discreteness effects one can distinguish between orbits which, in the continuum
limit, feel a regular potential, so that emittance grows as a power law in
time, and chaotic orbits, where emittance grows exponentially. For sufficiently
large N, one can distinguish two different `kinds' of chaos. Short range
microchaos, associated with close encounters between charges, is a generic
feature, yielding large positive Lyapunov exponents X_N which do not decrease
with increasing N even if the bulk potential is integrable. Alternatively,
there is the possibility of larger scale macrochaos, characterised by smaller
Lyapunov exponents X_S, which is present only if the bulk potential is chaotic.
Conventional computations of Lyapunov exponents probe X_N, leading to the
oxymoronic conclusion that N-body orbits which look nearly regular and have
sharply peaked Fourier spectra are `very chaotic.' However, the `range' of the
microchaos, set by the typical interparticle spacing, decreases as N increases,
so that, for large N, this microchaos, albeit very strong, is largely
irrelevant macroscopically. A more careful numerical analysis allows one to
estimate both X_N and X_S.Comment: 13 pages plus 17 figure

### Dynamics of triaxial galaxies with a central density cusp

Cuspy triaxial potentials admit a large number of chaotic orbits, which moreover exhibit extreme "stickiness" that makes the process of chaotic mixing surprisingly inefficient. Environmental effects, modeled as noise and/or periodic driving, help accelerate phase space transport but probably not as much as in simpler potentials. This could mean that cuspy triaxial ellipticals cannot exist as time-independent systems

### Chaos and the continuum limit in the gravitational N-body problem II. Nonintegrable potentials

This paper continues a numerical investigation of orbits evolved in `frozen,'
time-independent N-body realisations of smooth time-independent density
distributions corresponding to both integrable and nonintegrable potentials,
allowing for N as large as 300,000. The principal focus is on distinguishing
between, and quantifying, the effects of graininess on initial conditions
corresponding, in the continuum limit, to regular and chaotic orbits. Ordinary
Lyapunov exponents X do not provide a useful diagnostic for distinguishing
between regular and chaotic behaviour. Frozen-N orbits corresponding in the
continuum limit to both regular and chaotic characteristics have large positive
X even though, for large N, the `regular' frozen-N orbits closely resemble
regular characteristics in the smooth potential. Viewed macroscopically both
`regular' and `chaotic' frozen-N orbits diverge as a power law in time from
smooth orbits with the same initial condition. There is, however, an important
difference between `regular' and `chaotic' frozen-N orbits: For regular orbits,
the time scale associated with this divergence t_G ~ N^{1/2}t_D, with t_D a
characteristic dynamical time; for chaotic orbits t_G ~ (ln N) t_D. At least
for N>1000 or so, clear distinctions exist between phase mixing of initially
localised orbit ensembles which, in the continuum limit, exhibit regular versus
chaotic behaviour. For both regular and chaotic ensembles, finite-N effects are
well mimicked, both qualitatively and quantitatively, by energy-conserving
white noise with amplitude ~ 1/N. This suggests strongly that earlier
investigations of the effects of low amplitude noise on phase space transport
in smooth potentials are directly relevant to real physical systems.Comment: 20 pages, including 21 FIGURES, uses RevTeX macro

### Chaotic mixing in noisy Hamiltonian systems

This paper summarises an investigation of the effects of low amplitude noise
and periodic driving on phase space transport in 3-D Hamiltonian systems, a
problem directly applicable to systems like galaxies, where such perturbations
reflect internal irregularities and.or a surrounding environment. A new
diagnsotic tool is exploited to quantify how, over long times, different
segments of the same chaotic orbit 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
suprisingly short time scales. 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 in amplitude.
For white noise, the details are unimportant: additive and multiplicative noise
tend to have similar effects and the presence or absence of a 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 t here is little power at frequencies comparable
to 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.Comment: 15 pages including 18 figures, uses MNRAS LaTeX macro

### Chaos and Noise in a Truncated Toda Potential

Results are reported from a numerical investigation of orbits in a truncated
Toda potential which is perturbed by weak friction and noise. Two significant
conclusions are shown to emerge: (1) Despite other nontrivial behaviour,
configuration, velocity, and energy space moments associated with these
perturbations exhibit a simple scaling in the amplitude of the friction and
noise. (2) Even very weak friction and noise can induce an extrinsic diffusion
through cantori on a time scale much shorter than that associated with
intrinsic diffusion in the unperturbed system.Comment: 10 pages uuencoded PostScript (figures included), (A trivial
mathematical error leading to an erroneous conclusion is corrected

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