1,971 research outputs found
Stabilizing chaotic vortex trajectories: an example of high-dimensional control
A chaos control algorithm is developed to actively stabilize unstable
periodic orbits of higher-dimensional systems. The method assumes knowledge of
the model equations and a small number of experimentally accessible parameters.
General conditions for controllability are discussed. The algorithm is applied
to the Hamiltonian problem of point vortices inside a circular cylinder with
applications to an experimental plasma system.Comment: 15 LaTex pages, 4 Postscript figures adde
Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation
We show that Pyragas delayed feedback control can stabilize an unstable
periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of
a stable equilibrium in an n-dimensional dynamical system. This extends results
of Fiedler et al. [PRL 98, 114101 (2007)], who demonstrated that such feedback
control can stabilize the UPO associated with a two-dimensional subcritical
Hopf normal form. Pyragas feedback requires an appropriate choice of a feedback
gain matrix for stabilization, as well as knowledge of the period of the
targeted UPO. We apply feedback in the directions tangent to the
two-dimensional center manifold. We parameterize the feedback gain by a modulus
and a phase angle, and give explicit formulae for choosing these two parameters
given the period of the UPO in a neighborhood of the bifurcation point. We
show, first heuristically, and then rigorously by a center manifold reduction
for delay differential equations, that the stabilization mechanism involves a
highly degenerate Hopf bifurcation problem that is induced by the time-delayed
feedback. When the feedback gain modulus reaches a threshold for stabilization,
both of the genericity assumptions associated with a two-dimensional Hopf
bifurcation are violated: the eigenvalues of the linearized problem do not
cross the imaginary axis as the bifurcation parameter is varied, and the real
part of the cubic coefficient of the normal form vanishes. Our analysis of this
degenerate bifurcation problem reveals two qualitatively distinct cases when
unfolded in a two-parameter plane. In each case, Pyragas-type feedback
successfully stabilizes the branch of small-amplitude UPOs in a neighborhood of
the original bifurcation point, provided that the phase angle satisfies a
certain restriction.Comment: 35 pages, 19 figure
Classcial Bifurcation and Enhancement of Quantum Shells --- Systematic Analysis of Reflection-Asymmetric Deformed Oscillator ---
Correspondence between classical periodic orbits and quantum shell structure
is investigated for a reflection-asymmetric deformed oscillator model as a
function of quadrupole and octupole deformation parameters. Periodic orbit
theory reveals several aspects of quantum level structure for this
non-integrable system. Good classical- quantum correspondence is obtained in
the Fourier transform of the quantum level density, and importance of periodic
orbit bifurcation is demonstrated. Systematic survey of the local minima of
shell energies in the two-dimensional deformation parameter space shows that
prominent shell structures do emerge at finite values of the octupole
parameter. Correspondences between the regions exhibiting strong shell effects
and the classical bifurcation lines are investigated, and significance of these
bifurcations is indicated.Comment: 17 pages, REVTeX. 23 PostScript figures (not appended due to
excessive size, 3,860kb in total) are avalilable from K.A.
([email protected]) upon reques
Collinear helium under periodic driving: stabilization of the asymmetric stretch orbit
The collinear eZe configuration of helium, with the electrons on opposite
sides of the nucleus, is studied in the presence of an external electromagnetic
(laser or microwave) field. We show that the classically unstable "asymmetric
stretch" orbit, on which doubly excited intrashell states of helium with
maximum interelectronic angle are anchored, can be stabilized by means of a
resonant driving where the frequency of the electromagnetic field equals the
frequency of Kepler-like oscillations along the orbit. A static magnetic field,
oriented parallel to the oscillating electric field of the driving, can be used
to enforce the stability of the configuration with respect to deviations from
collinearity. Quantum Floquet calculations within a collinear model of the
driven two-electron atom reveal the existence of nondispersive wave packets
localized on the stabilized asymmetric stretch orbit, for double excitations
corresponding to principal quantum numbers of the order of N > 10.Comment: 13 pages, 12 figure
Existence of multi-site intrinsic localized modes in one-dimensional Debye crystals
The existence of highly localized multi-site oscillatory structures (discrete
multibreathers) in a nonlinear Klein-Gordon chain which is characterized by an
inverse dispersion law is proven and their linear stability is investigated.
The results are applied in the description of vertical (transverse, off-plane)
dust grain motion in dusty plasma crystals, by taking into account the lattice
discreteness and the sheath electric and/or magnetic field nonlinearity.
Explicit values from experimental plasma discharge experiments are considered.
The possibility for the occurrence of multibreathers associated with vertical
charged dust grain motion in strongly-coupled dusty plasmas (dust crystals) is
thus established. From a fundamental point of view, this study aims at
providing a first rigorous investigation of the existence of intrinsic
localized modes in Debye crystals and/or dusty plasma crystals and, in fact,
suggesting those lattices as model systems for the study of fundamental crystal
properties.Comment: 12 pages, 8 figures, revtex forma
Pattern Formation Induced by Time-Dependent Advection
We study pattern-forming instabilities in reaction-advection-diffusion
systems. We develop an approach based on Lyapunov-Bloch exponents to figure out
the impact of a spatially periodic mixing flow on the stability of a spatially
homogeneous state. We deal with the flows periodic in space that may have
arbitrary time dependence. We propose a discrete in time model, where reaction,
advection, and diffusion act as successive operators, and show that a mixing
advection can lead to a pattern-forming instability in a two-component system
where only one of the species is advected. Physically, this can be explained as
crossing a threshold of Turing instability due to effective increase of one of
the diffusion constants
Global attractors and extinction dynamics of cyclically competing species
Transitions to absorbing states are of fundamental importance in nonequilibrium physics as well as ecology. In ecology, absorbing states correspond to the extinction of species. We here study the spatial population dynamics of three cyclically interacting species. The interaction scheme comprises both direct competition between species as in the cyclic Lotka-Volterra model, and separated selection and reproduction processes as in the May-Leonard model. We show that the dynamic processes leading to the transient maintenance of biodiversity are closely linked to attractors of the nonlinear dynamics for the overall species' concentrations. The characteristics of these global attractors change qualitatively at certain threshold values of the mobility and depend on the relative strength of the different types of competition between species. They give information about the scaling of extinction times with the system size and thereby the stability of biodiversity. We define an effective free energy as the negative logarithm of the probability to find the system in a specific global state before reaching one of the absorbing states. The global attractors then correspond to minima of this effective energy landscape and determine the most probable values for the species' global concentrations. As in equilibrium thermodynamics, qualitative changes in the effective free energy landscape indicate and characterize the underlying nonequilibrium phase transitions. We provide the complete phase diagrams for the population dynamics and give a comprehensive analysis of the spatio-temporal dynamics and routes to extinction in the respective phases
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