142 research outputs found
Standing wave instabilities in a chain of nonlinear coupled oscillators
We consider existence and stability properties of nonlinear spatially
periodic or quasiperiodic standing waves (SWs) in one-dimensional lattices of
coupled anharmonic oscillators. Specifically, we consider Klein-Gordon (KG)
chains with either soft (e.g., Morse) or hard (e.g., quartic) on-site
potentials, as well as discrete nonlinear Schroedinger (DNLS) chains
approximating the small-amplitude dynamics of KG chains with weak inter-site
coupling. The SWs are constructed as exact time-periodic multibreather
solutions from the anticontinuous limit of uncoupled oscillators. In the
validity regime of the DNLS approximation these solutions can be continued into
the linear phonon band, where they merge into standard harmonic SWs. For SWs
with incommensurate wave vectors, this continuation is associated with an
inverse transition by breaking of analyticity. When the DNLS approximation is
not valid, the continuation may be interrupted by bifurcations associated with
resonances with higher harmonics of the SW. Concerning the stability, we
identify one class of SWs which are always linearly stable close to the
anticontinuous limit. However, approaching the linear limit all SWs with
nontrivial wave vectors become unstable through oscillatory instabilities,
persisting for arbitrarily small amplitudes in infinite lattices. Investigating
the dynamics resulting from these instabilities, we find two qualitatively
different regimes for wave vectors smaller than or larger than pi/2,
respectively. In one regime persisting breathers are found, while in the other
regime the system rapidly thermalizes.Comment: 57 pages, 21 figures, to be published in Physica D. Revised version:
Figs. 5 and 12 (f) replaced, some new results added to Sec. 5, Sec.7
(Conclusions) extended, 3 references adde
Global bifurcations to subcritical magnetorotational dynamo action in Keplerian shear flow
Magnetorotational dynamo action in Keplerian shear flow is a three-dimensional, non-linear magnetohydrodynamic process whose study is relevant to the understanding of accretion processes and magnetic field generation in astrophysics. Transition to this form of dynamo action is subcritical and shares many characteristics of transition to turbulence in non-rotating hydrodynamic shear flows. This suggests that these different fluid systems become active through similar generic bifurcation mechanisms, which in both cases have eluded detailed understanding so far. In this paper, we build on recent work on the two problems to investigate numerically the bifurcation mechanisms at work in the incompressible Keplerian magnetorotational dynamo problem in the shearing box framework. Using numerical techniques imported from dynamical systems research, we show that the onset of chaotic dynamo action at magnetic Prandtl numbers larger than unity is primarily associated with global homoclinic and heteroclinic bifurcations of nonlinear magnetorotational dynamo cycles. These global bifurcations are found to be supplemented by local bifurcations of cycles marking the beginning of period-doubling cascades. The results suggest that nonlinear magnetorotational dynamo cycles provide the pathway to turbulent injection of both kinetic and magnetic energy in incompressible magnetohydrodynamic Keplerian shear flow in the absence of an externally imposed magnetic field. Studying the nonlinear physics and bifurcations of these cycles in different regimes and configurations may subsequently help to better understand the physical conditions of excitation of magnetohydrodynamic turbulence and instability-driven dynamos in a variety of astrophysical systems and laboratory experiments. The detailed characterization of global bifurcations provided for this three-dimensional subcritical fluid dynamics problem may also prove useful for the problem of transition to turbulence in hydrodynamic shear flows
pde2path - A Matlab package for continuation and bifurcation in 2D elliptic systems
pde2path is a free and easy to use Matlab continuation/bifurcation package
for elliptic systems of PDEs with arbitrary many components, on general two
dimensional domains, and with rather general boundary conditions. The package
is based on the FEM of the Matlab pdetoolbox, and is explained by a number of
examples, including Bratu's problem, the Schnakenberg model, Rayleigh-Benard
convection, and von Karman plate equations. These serve as templates to study
new problems, for which the user has to provide, via Matlab function files, a
description of the geometry, the boundary conditions, the coefficients of the
PDE, and a rough initial guess of a solution. The basic algorithm is a one
parameter arclength continuation with optional bifurcation detection and
branch-switching. Stability calculations, error control and mesh-handling, and
some elementary time-integration for the associated parabolic problem are also
supported. The continuation, branch-switching, plotting etc are performed via
Matlab command-line function calls guided by the AUTO style. The software can
be downloaded from www.staff.uni-oldenburg.de/hannes.uecker/pde2path, where
also an online documentation of the software is provided such that in this
paper we focus more on the mathematics and the example systems
Statistical PT-symmetric lasing in an optical fiber network
PT-symmetry in optics is a condition whereby the real and imaginary parts of
the refractive index across a photonic structure are deliberately balanced.
This balance can lead to a host of novel optical phenomena, such as
unidirectional invisibility, loss-induced lasing, single-mode lasing from
multimode resonators, and non-reciprocal effects in conjunction with
nonlinearities. Because PT-symmetry has been thought of as fragile,
experimental realizations to date have been usually restricted to on-chip
micro-devices. Here, we demonstrate that certain features of PT-symmetry are
sufficiently robust to survive the statistical fluctuations associated with a
macroscopic optical cavity. We construct optical-fiber-based coupled-cavities
in excess of a kilometer in length (the free spectral range is less than 0.8
fm) with balanced gain and loss in two sub-cavities and examine the lasing
dynamics. In such a macroscopic system, fluctuations can lead to a
cavity-detuning exceeding the free spectral range. Nevertheless, by varying the
gain-loss contrast, we observe that both the lasing threshold and the growth of
the laser power follow the predicted behavior of a stable PT-symmetric
structure. Furthermore, a statistical symmetry-breaking point is observed upon
varying the cavity loss. These findings indicate that PT-symmetry is a more
robust optical phenomenon than previously expected, and points to potential
applications in optical fiber networks and fiber lasers.Comment: Submitted to Nature Communications, Pages 1-19: Main manuscript;
Pages 20-38: Supplementary material
Two-dimensional phase-space picture of the photonic crystal Fano laser
The recently realized photonic crystal Fano laser constitutes the first
demonstration of passive pulse generation in nanolasers [Nat. Photonics
, 81-84 (2017)]. We show that the laser operation is confined
to only two degrees-of-freedom after the initial transition stage. We show that
the original 5D dynamic model can be reduced to a 1D model in a narrow region
of the parameter space and it evolves into a 2D model after the exceptional
point, where the eigenvalues transition from being purely to a complex
conjugate pair. The 2D reduced model allows us to establish an effective band
structure for the eigenvalue problem of the stability matrix to explain the
laser dynamics. The reduced model is used to associate a previously unknown
origin of instability with a new unstable periodic orbit separating the stable
steady-state from the stable periodic orbit.Comment: 12 pages, 7 figures, journal, Phys. Rev. A, before editorial
correctio
Non-Hermitian Optics
From the viewpoint of quantum mechanics, a system must always be Hermitian since all its corresponding eigenvalues must be real. In contrast, the eigenvalues of open systems-unrestrained because of either decay or amplification-can be in general complex. Not so long ago, a certain class of non-Hermitian Hamiltonians was discovered that could have a completely real eigenvalue spectrum. This special class of Hamiltonians was found to respect the property of commutation with the parity-time (PT) operator. Translated into optics, this implies a balance between regions exhibiting gain and loss. Traditionally, loss has been perceived as a foe in optics and something that needs to be avoided at all costs. As we will show, when used in conjunction with gain, the presence of loss can lead to a host of counterintuitive outcomes in such non-Hermitian configurations that would have been otherwise unattainable in standard arrangements. We will study PT symmetric phase transitions in various optical settings that include semiconductor microrings and coupled fiber cavities, and show how they can allow mode-selectivity in lasers. One of the key outcomes of this effort was the realization of higher order degeneracies in a three-cavity laser configuration that can exhibit orders-of-magnitude larger sensitivity to external perturbations. We will also consider systems that display nonlinear effects such as gain saturation, thus allowing novel phase transitions. Some interesting properties associated with degeneracies in non-Hermitian settings will be investigated as well. Such degeneracies, called exceptional points (EPs), are much more drastic compared to standard degeneracies of eigenvalues because the corresponding eigenvectors also coalesce, which in turn reduces the dimensionality of the phase space. We will show that dynamic parameter contours enclosing or close to EPs can lead to a robust chiral mode conversion process – something that can be potentially used to realize omni-polarizing optical devices
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
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