5,607 research outputs found
Frequency Locking in Spatially Extended Systems
A variant of the complex Ginzburg-Landau equation is used to investigate the
frequency locking phenomena in spatially extended systems. With appropriate
parameter values, a variety of frequency-locked patterns including flats,
fronts, labyrinths and fronts emerge. We show that in spatially
extended systems, frequency locking can be enhanced or suppressed by diffusive
coupling. Novel patterns such as chaotically bursting domains and target
patterns are also observed during the transition to locking
Four-phase patterns in forced oscillatory systems
We investigate pattern formation in self-oscillating systems forced by an
external periodic perturbation. Experimental observations and numerical studies
of reaction-diffusion systems and an analysis of an amplitude equation are
presented. The oscillations in each of these systems entrain to rational
multiples of the perturbation frequency for certain values of the forcing
frequency and amplitude. We focus on the subharmonic resonant case where the
system locks at one fourth the driving frequency, and four-phase rotating
spiral patterns are observed at low forcing amplitudes. The spiral patterns are
studied using an amplitude equation for periodically forced oscillating
systems. The analysis predicts a bifurcation (with increasing forcing) from
rotating four-phase spirals to standing two-phase patterns. This bifurcation is
also found in periodically forced reaction-diffusion equations, the
FitzHugh-Nagumo and Brusselator models, even far from the onset of oscillations
where the amplitude equation analysis is not strictly valid. In a
Belousov-Zhabotinsky chemical system periodically forced with light we also
observe four-phase rotating spiral wave patterns. However, we have not observed
the transition to standing two-phase patterns, possibly because with increasing
light intensity the reaction kinetics become excitable rather than oscillatory.Comment: 11 page
Unusual synchronization phenomena during electrodissolution of silicon: the role of nonlinear global coupling
The photoelectrodissolution of n-type silicon constitutes a convenient model
system to study the nonlinear dynamics of oscillatory media. On the silicon
surface, a silicon oxide layer forms. In the lateral direction, the thickness
of this layer is not uniform. Rather, several spatio-temporal patterns in the
oxide layer emerge spontaneously, ranging from cluster patterns and turbulence
to quite peculiar dynamics like chimera states. Introducing a nonlinear global
coupling in the complex Ginzburg-Landau equation allows us to identify this
nonlinear coupling as the essential ingredient to describe the patterns found
in the experiments. The nonlinear global coupling is designed in such a way, as
to capture an important, experimentally observed feature: the spatially
averaged oxide-layer thickness shows nearly harmonic oscillations. Simulations
of the modified complex Ginzburg-Landau equation capture the experimental
dynamics very well.Comment: To appear as a chapter in "Engineering of Chemical Complexity II"
(eds. A.S. Mikhailov and G.Ertl) at World Scientific in Singapor
Bose-Einstein Condensates in Superlattices
We consider the Gross--Pitaevskii (GP) equation in the presence of periodic and quasi-periodic superlattices to study cigar-shaped Bose--Einstein condensates (BECs) in such potentials. We examine spatially extended wavefunctions in the form of modulated amplitude waves (MAWs). With a coherent structure ansatz, we derive amplitude equations describing the evolution of spatially modulated states of the BEC. We then apply second-order multiple scale perturbation theory to study harmonic resonances with respect to a single lattice substructure as well as ultrasubharmonic resonances that result from interactions of both substructures of the superlattice. In each case, we determine the resulting system's equilibria, which represent spatially periodic solutions, and subsequently examine the stability of the corresponding wavefunctions by direct simulations of the GP equation, identifying them as typically stable solutions of the model. We then study subharmonic resonances using Hamiltonian perturbation theory, tracing robust spatio-temporally periodic patterns
Cooperative surmounting of bottlenecks
The physics of activated escape of objects out of a metastable state plays a
key role in diverse scientific areas involving chemical kinetics, diffusion and
dislocation motion in solids, nucleation, electrical transport, motion of flux
lines superconductors, charge density waves, and transport processes of
macromolecules, to name but a few. The underlying activated processes present
the multidimensional extension of the Kramers problem of a single Brownian
particle. In comparison to the latter case, however, the dynamics ensuing from
the interactions of many coupled units can lead to intriguing novel phenomena
that are not present when only a single degree of freedom is involved. In this
review we report on a variety of such phenomena that are exhibited by systems
consisting of chains of interacting units in the presence of potential
barriers.
In the first part we consider recent developments in the case of a
deterministic dynamics driving cooperative escape processes of coupled
nonlinear units out of metastable states. The ability of chains of coupled
units to undergo spontaneous conformational transitions can lead to a
self-organised escape. The mechanism at work is that the energies of the units
become re-arranged, while keeping the total energy conserved, in forming
localised energy modes that in turn trigger the cooperative escape. We present
scenarios of significantly enhanced noise-free escape rates if compared to the
noise-assisted case.
The second part deals with the collective directed transport of systems of
interacting particles overcoming energetic barriers in periodic potential
landscapes. Escape processes in both time-homogeneous and time-dependent driven
systems are considered for the emergence of directed motion. It is shown that
ballistic channels immersed in the associated high-dimensional phase space are
the source for the directed long-range transport
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