60 research outputs found
Nonintegrable Schrodinger Discrete Breathers
In an extensive numerical investigation of nonintegrable translational motion
of discrete breathers in nonlinear Schrodinger lattices, we have used a
regularized Newton algorithm to continue these solutions from the limit of the
integrable Ablowitz-Ladik lattice. These solutions are shown to be a
superposition of a localized moving core and an excited extended state
(background) to which the localized moving pulse is spatially asymptotic. The
background is a linear combination of small amplitude nonlinear resonant plane
waves and it plays an essential role in the energy balance governing the
translational motion of the localized core. Perturbative collective variable
theory predictions are critically analyzed in the light of the numerical
results.Comment: 42 pages, 28 figures. to be published in CHAOS (December 2004
TIME-DEPENDENT SYSTEMS AND CHAOS IN STRING THEORY
One of the phenomenal results emerging from string theory is the AdS/CFT correspondence or gauge-gravity duality: In certain cases a theory of gravity is equivalent to a dual gauge theory, very similar to the one describing non-gravitational interactions of fundamental subatomic particles. A difficult problem on one side can be mapped to a simpler and solvable problem on the other side using this correspondence. Thus one of the theories can be understood better using the other.
The mapping between theories of gravity and gauge theories has led to new approaches to building models of particle physics from string theory. One of the important features to model is the phenomenon of confinement present in strong interaction of particle physics. This feature is not present in the gauge theory arising in the simplest of the examples of the duality. However this N = 4 supersymmetric Yang-Mills gauge theory enjoys the property of being integrable, i.e. it can be exactly solved in terms of conserved charges. It is expected that if a more realistic theory turns out to be integrable, solvability of the theory would lead to simple analytical expressions for quantities like masses of the hadrons in the theory. In this thesis we show that the existing models of confinement are all nonintegrable--such simple analytic expressions cannot be obtained.
We moreover show that these nonintegrable systems also exhibit features of chaotic dynamical systems, namely, sensitivity to initial conditions and a typical route of transition to chaos. We proceed to study the quantum mechanics of these systems and check whether their properties match those of chaotic quantum systems. Interestingly, the distribution of the spacing of meson excitations measured in the laboratory have been found to match with level-spacing distribution of typical quantum chaotic systems. We find agreement of this distribution with models of confining strong interactions, conforming these as viable models of particle physics arising from string theory
Chaos in cosmological Hamiltonians
This paper summarises a numerical investigation which aimed to identify and
characterise regular and chaotic behaviour in time-dependent Hamiltonians
H(r,p,t) = p^2/2 + U(r,t), with U=R(t)V(r) or U=V[R(t)r], where V(r) is a
polynomial in x, y, and/or z, and R = const * t^p is a time-dependent scale
factor. When p is not too negative, one can distinguish between regular and
chaotic behaviour by determining whether an orbit segment exhibits a sensitive
dependence on initial conditions. However, chaotic segments in these potentials
differ from chaotic segments in time-independent potentials in that a small
initial perturbation will usually exhibit a sub- or super-exponential growth in
time. Although not periodic, regular segments typically exhibit simpler shapes,
topologies, and Fourier spectra than do chaotic segments. This distinction
between regular and chaotic behaviour is not absolute since a single orbit
segment can seemingly change from regular to chaotic and visa versa. All these
observed phenomena can be understood in terms of a simple theoretical model.Comment: 16 pages LaTeX, including 5 figures, no macros require
Homoclinic chaos in the dynamics of a general Bianchi IX model
The dynamics of a general Bianchi IX model with three scale factors is
examined. The matter content of the model is assumed to be comoving dust plus a
positive cosmological constant. The model presents a critical point of
saddle-center-center type in the finite region of phase space. This critical
point engenders in the phase space dynamics the topology of stable and unstable
four dimensional tubes , where is a saddle direction and
is the manifold of unstable periodic orbits in the center-center sector.
A general characteristic of the dynamical flow is an oscillatory mode about
orbits of an invariant plane of the dynamics which contains the critical point
and a Friedmann-Robertson-Walker (FRW) singularity. We show that a pair of
tubes (one stable, one unstable) emerging from the neighborhood of the critical
point towards the FRW singularity have homoclinic transversal crossings. The
homoclinic intersection manifold has topology and is constituted
of homoclinic orbits which are bi-asymptotic to the center-center
manifold. This is an invariant signature of chaos in the model, and produces
chaotic sets in phase space. The model also presents an asymptotic DeSitter
attractor at infinity and initial conditions sets are shown to have fractal
basin boundaries connected to the escape into the DeSitter configuration
(escape into inflation), characterizing the critical point as a chaotic
scatterer.Comment: 11 pages, 6 ps figures. Accepted for publication in Phys. Rev.
Classical quasi-particle dynamics in trapped Bose condensates
The dynamics of quasi-particles in repulsive Bose condensates in a harmonic
trap is studied in the classical limit. In isotropic traps the classical motion
is integrable and separable in spherical coordinates. In anisotropic traps the
classical dynamics is found, in general, to be nonintegrable. For
quasi-particle energies E much smaller than thechemical potential, besides the
conserved quasi-particle energy, we identify two additional nearly conserved
phase-space functions. These render the dynamics inside the condensate
(collective dynamics) integrable asymptotically for E/chemical potential very
small. However, there coexists at the same energy a dynamics confined to the
surface of the condensate, which is governed by a classical Hartree-Fock
Hamiltonian. We find that also this dynamics becomes integrable for E/chemical
potential very small, because of the appearance of an adiabatic invariant. For
E/chemical potential of order 1 a large portion of the phase-space supports
chaotic motion, both, for the Bogoliubov Hamiltonian and its Hartree-Fock
approximant. To exemplify this we exhibit Poincar\'e surface of sections for
harmonic traps with the cylindrical symmetry and anisotropy found in TOP traps.
For E/chemical potential very large the dynamics is again governed by the
Hartree-Fock Hamiltonian. In the case with cylindrical symmetry it becomes
quasi-integrable because the remaining small chaotic components in phase space
are tightly confined by tori.Comment: 13 pages Latex, 6 eps.gz-figure
Slowly-rotating compact objects: the nonintegrability of Hartle-Thorne particle geodesics
X-ray astronomy provides information regarding the electromagnetic emission
of active galactic nuclei and X-ray binaries. These events provide details
regarding the astrophysical environment of black holes and stars, and help us
understand gamma-ray bursts. They produce estimates for the maximum mass of
neutron stars and eventually will contribute to the discovery of their equation
of state. Thus, it is crucial to study these configurations to increase the
yield of X-ray astronomy when combined with multimessenger gravitational-wave
astrophysics and black hole shadows. Unfortunately, an exact solution of the
field equations does not exist for neutron stars. Nevertheless, there exist a
variety of approximate compact objects that may characterize massive or neutron
stars. The most studied approximation is the Hartle-Thorne metric that
represents slowly-rotating compact objects, like massive stars, white dwarfs
and neutron stars. Recent investigations of photon orbits and shadows of such
metric revealed that it exhibits chaos close to resonances. Here, we thoroughly
investigate particle orbits around the Hartle-Thorne spacetime. We perform an
exhaustive analysis of bound motion, by varying all parameters involved in the
system. We demonstrate that chaotic regions, known as Birkhoff islands, form
around resonances, where the ratio of the radial and polar frequency of
geodesics, known as the rotation number, is shared throughout the island. This
leads to the formation of plateaus in rotation curves during the most prominent
resonance, which designate nonintegrability. We measure their width and
show how each parameter affects it. The nonintegrability of Hartle-Thorne
metric may affect quasiperiodic oscillations of low-mass X-ray binaries, when
chaos is taken into account, and improve estimates of mass, angular momentum
and multipole moments of astrophysical compact objects.Comment: 12 pages, 5 figure
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