8 research outputs found
A Weakly Nonlinear Analysis of Impulsively-Forced Faraday Waves
Parametrically-excited surface waves, forced by a periodic sequence of
delta-function impulses, are considered within the framework of the
Zhang-Vi\~nals model (J. Fluid Mech. 1997). The exact impulsive-forcing
results, in the linear and weakly nonlinear regimes, are compared with
numerical results for sinusoidal and multifrequency forcing. We find
surprisingly good agreement between impulsive forcing results and those
obtained using a two-term truncated Fourier series representation of the
impulsive forcing function. As noted previously by Bechhoefer and Johnson (Am.
J. Phys. 1996), in the case of two equally-spaced impulses per period there are
only subharmonic modes of instability. The familiar situation of alternating
subharmonic and harmonic resonance tongues emerges for unequally-spaced
impulses. We extend the linear analysis for two impulses per period to the
weakly nonlinear regime for one-dimensional waves. Specifically, we derive an
analytic expression for the cubic Landau coefficient in the bifurcation
equation as a function of the dimensionless fluid parameters and spacing
between the two impulses. As the capillary parameter is varied, one finds a
parameter region of wave amplitude suppression, which is due to a familiar 1:2
spatio-temporal resonance between the subharmonic mode of instability and a
damped harmonic mode. This resonance occurs for impulsive forcing even when
harmonic resonance tongues are absent from the neutral stability curve. The
strength of this resonance feature can be tuned by varying the spacing between
the impulses. This finding is interpreted in terms of a recent symmetry-based
analysis of multifrequency forced Faraday waves by Porter, Topaz and Silber
(Phys. Rev. Lett. 2004, Phys. Rev. E 2004).Comment: 13 pages, 10 figures, submitted to Physical Review
Forced patterns near a Turing-Hopf bifurcation
We study time-periodic forcing of spatially-extended patterns near a
Turing-Hopf bifurcation point. A symmetry-based normal form analysis yields
several predictions, including that (i) weak forcing near the intrinsic Hopf
frequency enhances or suppresses the Turing amplitude by an amount that scales
quadratically with the forcing strength, and (ii) the strongest effect is seen
for forcing that is detuned from the Hopf frequency. To apply our results to
specific models, we perform a perturbation analysis on general two-component
reaction-diffusion systems, which reveals whether the forcing suppresses or
enhances the spatial pattern. For the suppressing case, our results explain
features of previous experiments on the CDIMA chemical reaction. However, we
also find examples of the enhancing case, which has not yet been observed in
experiment. Numerical simulations verify the predicted dependence on the
forcing parameters.Comment: 4 pages, 4 figure
limit solutions for control systems
For a control Cauchy problem on an interval , we
propose a notion of limit solution verifying the following properties: i)
is defined for (impulsive) inputs and for standard,
bounded measurable, controls ; ii) in the commutative case (i.e. when
for all ),
coincides with the solution one can obtain via the change of coordinates that
makes the simultaneously constant; iii) subsumes former concepts
of solution valid for the generic, noncommutative case.
In particular, when has bounded variation, we investigate the relation
between limit solutions and (single-valued) graph completion solutions.
Furthermore, we prove consistency with the classical Carath\'eodory solution
when and are absolutely continuous.
Even though some specific problems are better addressed by means of special
representations of the solutions, we believe that various theoretical issues
call for a unified notion of trajectory. For instance, this is the case of
optimal control problems, possibly with state and endpoint constraints, for
which no extra assumptions (like e.g. coercivity, bounded variation,
commutativity) are made in advance