2,256 research outputs found
Control of Dynamic Hopf Bifurcations
The slow passage through a Hopf bifurcation leads to the delayed appearance
of large amplitude oscillations. We construct a smooth scalar feedback control
which suppresses the delay and causes the system to follow a stable equilibrium
branch. This feature can be used to detect in time the loss of stability of an
ageing device. As a by-product, we obtain results on the slow passage through a
bifurcation with double zero eigenvalue, described by a singularly perturbed
cubic Lienard equation.Comment: 25 pages, 4 figure
Delayed feedback control of self-mobile cavity solitons in a wide-aperture laser with a saturable absorber
We investigate the spatiotemporal dynamics of cavity solitons in a broad area
vertical-cavity surface-emitting laser with saturable absorption subjected to
time-delayed optical feedback. Using a combination of analytical, numerical and
path continuation methods we analyze the bifurcation structure of stationary
and moving cavity solitons and identify two different types of traveling
localized solutions, corresponding to slow and fast motion. We show that the
delay impacts both stationary and moving solutions either causing drifting and
wiggling dynamics of initially stationary cavity solitons or leading to
stabilization of intrinsically moving solutions. Finally, we demonstrate that
the fast cavity solitons can be associated with a lateral mode-locking regime
in a broad-area laser with a single longitudinal mode
Geometric stabilization of extended S=2 vortices in two-dimensional photonic lattices: theoretical analysis, numerical computation and experimental results
In this work, we focus our studies on the subject of nonlinear discrete
self-trapping of S=2 (doubly-charged) vortices in two-dimensional photonic
lattices, including theoretical analysis, numerical computation and
experimental demonstration. We revisit earlier findings about S=2 vortices with
a discrete model, and find that S=2 vortices extended over eight lattice sites
can indeed be stable (or only weakly unstable) under certain conditions, not
only for the cubic nonlinearity previously used, but also for a saturable
nonlinearity more relevant to our experiment with a biased photorefractive
nonlinear crystal. We then use the discrete analysis as a guide towards
numerically identifying stable (and unstable) vortex solutions in a more
realistic continuum model with a periodic potential. Finally, we present our
experimental observation of such geometrically extended S=2 vortex solitons in
optically induced lattices under both self-focusing and self-defocusing
nonlinearities, and show clearly that the S=2 vortex singularities are
preserved during nonlinear propagation
Revisiting linear augmentation for stabilizing stationary solutions: potential pitfalls and their application
Linear augmentation has recently been shown to be effective in targeting
desired stationary solutions, suppressing bistablity, in regulating the
dynamics of drive response systems and in controlling the dynamics of hidden
attractors. The simplicity of the procedure is the highlight of this scheme but
at the same time questions related to its general applicability still need to
be addressed. Focusing on the issue of targeting stationary solutions, this
work demonstrates instances where the scheme fails to stabilize the required
solutions and leads to other complicated dynamical scenarios. Appropriate
examples from conservative as well as dissipative systems are presented in this
regard and potential applications for relevant observations in dissipative
predator--prey systems are also discussed.Comment: updated version with title change, additional figures, text and
explanation
Solitons in a parametrically driven damped discrete nonlinear Schr\"odinger equation
We consider a parametrically driven damped discrete nonlinear Schr\"odinger
(PDDNLS) equation. Analytical and numerical calculations are performed to
determine the existence and stability of fundamental discrete bright solitons.
We show that there are two types of onsite discrete soliton, namely onsite type
I and II. We also show that there are four types of intersite discrete soliton,
called intersite type I, II, III, and IV, where the last two types are
essentially the same, due to symmetry. Onsite and intersite type I solitons,
which can be unstable in the case of no dissipation, are found to be stabilized
by the damping, whereas the other types are always unstable. Our further
analysis demonstrates that saddle-node and pitchfork (symmetry-breaking)
bifurcations can occur. More interestingly, the onsite type I, intersite type
I, and intersite type III-IV admit Hopf bifurcations from which emerge periodic
solitons (limit cycles). The continuation of the limit cycles as well as the
stability of the periodic solitons are computed through the numerical
continuation software Matcont. We observe subcritical Hopf bifurcations along
the existence curve of the onsite type I and intersite type III-IV. Along the
existence curve of the intersite type I we observe both supercritical and
subcritical Hopf bifurcations.Comment: to appear in "Spontaneous Symmetry Breaking, Self-Trapping, and
Josephson Oscillations in Nonlinear Systems", B.A. Malomed, ed. (Springer,
Berlin, 2012
Clustering of exceptional points and dynamical phase transitions
The eigenvalues of a non-Hermitian Hamilton operator are complex and provide
not only the energies but also the lifetimes of the states of the system. They
show a non-analytical behavior at singular (exceptional) points (EPs). The
eigenfunctions are biorthogonal, in contrast to the orthogonal eigenfunctions
of a Hermitian operator. A quantitative measure for the ratio between
biorthogonality and orthogonality is the phase rigidity of the wavefunctions.
At and near an EP, the phase rigidity takes its minimum value. The lifetimes of
two nearby eigenstates of a quantum system bifurcate under the influence of an
EP. When the parameters are tuned to the point of maximum width bifurcation,
the phase rigidity suddenly increases up to its maximum value. This means that
the eigenfunctions become almost orthogonal at this point. This unexpected
result is very robust as shown by numerical results for different classes of
systems. Physically, it causes an irreversible stabilization of the system by
creating local structures that can be described well by a Hermitian Hamilton
operator. Interesting non-trivial features of open quantum systems appear in
the parameter range in which a clustering of EPs causes a dynamical phase
transition.Comment: A few improvements; 2 references added; 28 pages; 7 figure
Symmetry-breaking transitions in networks of nonlinear circuit elements
We investigate a nonlinear circuit consisting of N tunnel diodes in series,
which shows close similarities to a semiconductor superlattice or to a neural
network. Each tunnel diode is modeled by a three-variable FitzHugh-Nagumo-like
system. The tunnel diodes are coupled globally through a load resistor. We find
complex bifurcation scenarios with symmetry-breaking transitions that generate
multiple fixed points off the synchronization manifold. We show that multiply
degenerate zero-eigenvalue bifurcations occur, which lead to multistable
current branches, and that these bifurcations are also degenerate with a Hopf
bifurcation. These predicted scenarios of multiple branches and degenerate
bifurcations are also found experimentally.Comment: 32 pages, 11 figures, 7 movies available as ancillary file
- …