645 research outputs found
The Electronic Spectrum of Fullerenes from the Dirac Equation
The electronic spectrum of sheets of graphite (plane honeycomb lattice)
folded into regular polihedra is studied. A continuum limit valid for
sufficiently large molecules and based on a tight binding approximation is
derived. It is found that a Dirac equation describes the flat graphite lattice.
Curving the lattice by insertion of odd numbered rings can be mimicked by
coupling effective gauge fields. In particular the and related
molecules are well described by the Dirac equation on the surface of a sphere
coupled to a color monopole sitting at its center.Comment: 29 pages, 7 figures. IASSNS-HEP-92/5
Multi-black holes from nilpotent Lie algebra orbits
For N \ge 2 supergravities, BPS black hole solutions preserving four
supersymmetries can be superposed linearly, leading to well defined solutions
containing an arbitrary number of such BPS black holes at arbitrary positions.
Being stationary, these solutions can be understood via associated non-linear
sigma models over pseudo-Riemaniann spaces coupled to Euclidean gravity in
three spatial dimensions. As the main result of this paper, we show that
whenever this pseudo-Riemanniann space is an irreducible symmetric space G/H*,
the most general solutions of this type can be entirely characterised and
derived from the nilpotent orbits of the associated Lie algebra Lie(G). This
technique also permits the explicit computation of non-supersymmetric extremal
solutions which cannot be obtained by truncation to N=2 supergravity theories.
For maximal supergravity, we not only recover the known BPS solutions depending
on 32 independent harmonic functions, but in addition find a set of non-BPS
solutions depending on 29 harmonic functions. While the BPS solutions can be
understood within the appropriate N=2 truncation of N=8 supergravity, the
general non-BPS solutions require the whole field content of the theory.Comment: Corrected version for publication, references adde
Spontaneous synchronization of two bistable pyridine-furan nanosprings connected by an oligomeric bridge
The intensive development of nanodevices acting as two-state systems has
motivated the search for nanoscale molecular structures whose long-term
conformational dynamics are similar to the dynamics of bistable mechanical
systems such as Euler arches and Duffing oscillators. Collective synchrony in
bistable dynamics of molecular-sized systems has attracted immense attention as
a potential pathway to amplify the output signals of molecular nanodevices.
Recently, pyridin-furan oligomers of helical shape that are a few nanometers in
size and exhibit bistable dynamics similar to a Duffing oscillator have been
identified through molecular dynamics simulations. In this article, we present
the case of dynamical synchronization of these bistable systems. We show that
two pyridine-furan springs connected by a rigid oligomeric bridge spontaneously
synchronize vibrations and stochastic resonance enhances the synchronization
effect
Applications of dynamical systems with symmetry
This thesis examines the application of symmetric dynamical systems theory to
two areas in applied mathematics: weakly coupled oscillators with symmetry, and
bifurcations in flame front equations.
After a general introduction in the first chapter, chapter 2 develops a theoretical
framework for the study of identical oscillators with arbitrary symmetry group under an
assumption of weak coupling. It focusses on networks with 'all to all' Sn coupling. The
structure imposed by the symmetry on the phase space for weakly coupled oscillators
with Sn, Zn or Dn symmetries is discussed, and the interaction of internal symmetries
and network symmetries is shown to cause decoupling under certain conditions.
Chapter 3 discusses what this implies for generic dynamical behaviour of coupled
oscillator systems, and concentrates on application to small numbers of oscillators (three
or four). We find strong restrictions on bifurcations, and structurally stable heteroclinic
cycles.
Following this, chapter 4 reports on experimental results from electronic oscillator
systems and relates it to results in chapter 3. In a forced oscillator system, breakdown
of regular motion is observed to occur through break up of tori followed by a symmetric
bifurcation of chaotic attractors to fully symmetric chaos.
Chapter 5 discusses reduction of a system of identical coupled oscillators to phase
equations in a weakly coupled limit, considering them as weakly dissipative Hamiltonian
oscillators with very weakly coupling. This provides a derivation of example phase
equations discussed in chapter 2. Applications are shown for two van der Pol-Duffing
oscillators in the case of a twin-well potential.
Finally, we turn our attention to the Kuramoto-Sivashinsky equation. Chapter 6
starts by discussing flame front equations in general, and non-linear models in particular.
The Kuramoto-Sivashinsky equation on a rectangular domain with simple
boundary conditions is found to be an example of a large class of systems whose linear
behaviour gives rise to arbitrarily high order mode interactions.
Chapter 7 presents computation of some of these mode interactions using competerised
Liapunov-Schmidt reduction onto the kernel of the linearisation, and investigates
the bifurcation diagrams in two parameters
The "Coulomb phase" in frustrated systems
The "Coulomb phase" is an emergent state for lattice models (particularly
highly frustrated antiferromagnets) which have local constraints that can be
mapped to a divergence-free "flux". The coarse-grained version of this flux or
polarization behave analogously to electric or magnetic fields; in particular,
defects at which the local constraint is violated behave as effective charges
with Coulomb interactions. I survey the derivation of the characteristic
power-law correlation functions and the pinch-points in reciprocal space plots
of diffuse scattering, as well as applications to magnetic relaxation,
quantum-mechanical generalizations, phase transitions to long-range-ordered
states, and the effects of disorder.Comment: 30 pp, 5 figures (Sub. to Annual Reviews of Condensed Matter Physics
Dynamics and Synchronization of Weak Chimera States for a Coupled Oscillator System
This thesis is an investigation of chimera states in a network of identical coupled phase
oscillators. Chimera states are intriguing phenomena that can occur in systems of coupled
identical phase oscillators when synchronized and desynchronized oscillators coexist.
We use the Kuramoto model and coupling function of Hansel for a specific system of six
oscillators to prove the existence of chimera states.
More precisely, we prove analytically there are chimera states in a small network of
six phase oscillators previously investigated numerically by Ashwin and Burylko [8]. We
can reduce to a two-dimensional system within an invariant subspace, in terms of phase
differences. This system is found to have an integral of motion for a specific choice of
parameters. Using this we prove there is a set of periodic orbits that is a weak chimera.
Moreover, we are able to confirmthat there is an infinite number of chimera states at the
special case of parameters, using the weak chimera definition of [8].
We approximate the Poincaré return map for these weak chimera solutions and demonstrate
several results about their stability and bifurcation for nearby parameters. These agree
with numerical path following of the solutions.
We also consider another invariant subspace to reduce the Kuramoto model of six
coupled phase oscillators to a first order differential equation. We analyse this equation
numerically and find regions of attracting chimera states exist within this invariant subspace.
By computing eigenvalues at a nonhyperbolic point for the system of phase differences, we
numerically find there are chimera states in the invariant subspace that are attracting within
full system.Republic of Iraq,
Ministry of Higher Education and Scientific Research
Symmetry breaking yields chimeras in two small populations of Kuramoto-type oscillators
Despite their simplicity, networks of coupled phase oscillators can give rise
to intriguing collective dynamical phenomena. However, the symmetries of
globally and identically coupled identical units do not allow solutions where
distinct oscillators are frequency-unlocked -- a necessary condition for the
emergence of chimeras. Thus, forced symmetry breaking is necessary to observe
chimera-type solutions. Here, we consider the bifurcations that arise when full
permutational symmetry is broken for the network to consist of coupled
populations. We consider the smallest possible network composed of four phase
oscillators and elucidate the phase space structure, (partial) integrability
for some parameter values, and how the bifurcations away from full symmetry
lead to frequency-unlocked weak chimera solutions. Since such solutions wind
around a torus they must arise in a global bifurcation scenario. Moreover,
periodic weak chimeras undergo a period doubling cascade leading to chaos. The
resulting chaotic dynamics with distinct frequencies do not rely on amplitude
variation and arise in the smallest networks that support chaos
Design of a YIG-tuned oscillator
A technique for designing YIG (yttrium-iron garnet) tuned transistor oscillators, tunable over the range of frequencies from 500 to 950 MHZ, is presented. The approach taken differs appreciably from that used in the design of conventional LC-tuned oscillators. One major difference is that the YIG tuning mechanism is electrically controlled. The YIG tuning element is treated as a single unit and is not resolved into an equivalent LC circuit. Instead, a direct method using reflection coefficients measured at network terminals to characterize various design stages is applied. The transistor is also characterized by reflection and transmission coefficients, i.e., S-parameters. Thus the Smith Chart becomes a useful tool and network calculations are greatly simplified by means of signal flow analysis with application of Mason\u27s rule. Because S-parameters are measured when the device is terminated in the characteristic impedance of the measuring system, they are more accurately determined at high frequencies than other parameters requiring open and short circuit terminations for their measurement. Furthermore, the availability of network analyzers, such as the Hewlett-Packard S-Parameter Test Set, simplifies such measurements. As a result, a concise method using S-parameters is most applicable for the design of transistor YIG-tuned oscillators
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