221 research outputs found
On the state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain
The continuous and discrete symmetries of the Kuramoto-Sivashinsky system
restricted to a spatially periodic domain play a prominent role in shaping the
invariant sets of its chaotic dynamics. The continuous spatial translation
symmetry leads to relative equilibrium (traveling wave) and relative periodic
orbit (modulated traveling wave) solutions. The discrete symmetries lead to
existence of equilibrium and periodic orbit solutions, induce decomposition of
state space into invariant subspaces, and enforce certain structurally stable
heteroclinic connections between equilibria. We show, on the example of a
particular small-cell Kuramoto-Sivashinsky system, how the geometry of its
dynamical state space is organized by a rigid `cage' built by heteroclinic
connections between equilibria, and demonstrate the preponderance of unstable
relative periodic orbits and their likely role as the skeleton underpinning
spatiotemporal turbulence in systems with continuous symmetries. We also offer
novel visualizations of the high-dimensional Kuramoto-Sivashinsky state space
flow through projections onto low-dimensional, PDE representation independent,
dynamically invariant intrinsic coordinate frames, as well as in terms of the
physical, symmetry invariant energy transfer rates.Comment: 31 pages, 17 figures; added references, corrected typos. Due to file
size restrictions some figures in this preprint are of low quality. A high
quality copy may be obtained from
http://www.cns.gatech.edu/~predrag/papers/preprints.html#rp
Rigorous derivation of the Kuramoto-Sivashinsky equation in a 2D weakly nonlinear Stefan problem
In this paper we are interested in a rigorous derivation of the
Kuramoto-Sivashinsky equation (K--S) in a Free Boundary Problem. As a paradigm,
we consider a two-dimensional Stefan problem in a strip, a simplified version
of a solid-liquid interface model. Near the instability threshold, we introduce
a small parameter and define rescaled variables accordingly. At
fixed , our method is based on: definition of a suitable linear 1D
operator, projection with respect to the longitudinal coordinate only,
Lyapunov-Schmidt method. As a solvability condition, we derive a
self-consistent parabolic equation for the front. We prove that, starting from
the same configuration, the latter remains close to the solution of K--S on a
fixed time interval, uniformly in sufficiently small
Stabilized Kuramoto-Sivashinsky equation: A useful model for secondary instabilities and related dynamics of experimental one-dimensional cellular flows
We report numerical simulations of one-dimensional cellular solutions of the
stabilized Kuramoto-Sivashinsky equation. This equation offers a range of
generic behavior in pattern-forming instabilities of moving interfaces, such as
a host of secondary instabilities or transition toward disorder. We compare
some of these collective behaviors to those observed in experiments. In
particular, destabilization scenarios of bifurcated states are studied in a
spatially semi-extended situation, which is common in realistic patterns, but
has been barely explored so far.Comment: 4 pages, 14 figure
Spatio-temporal dynamics of an active, polar, viscoelastic ring
Constitutive equations for a one-dimensional, active, polar, viscoelastic
liquid are derived by treating the strain field as a slow hydrodynamic
variable. Taking into account the couplings between strain and polarity allowed
by symmetry, the hydrodynamics of an active, polar, viscoelastic body include
an evolution equation for the polarity field that generalizes the damped
Kuramoto-Sivashinsky equation. Beyond thresholds of the active coupling
coefficients between the polarity and the stress or the strain rate,
bifurcations of the homogeneous state lead first to stationary waves, then to
propagating waves of the strain, stress and polarity fields. I argue that these
results are relevant to living matter, and may explain rotating actomyosin
rings in cells and mechanical waves in epithelial cell monolayers.Comment: 9 pages, 4 figure
Microextensive Chaos of a Spatially Extended System
By analyzing chaotic states of the one-dimensional Kuramoto-Sivashinsky
equation for system sizes L in the range 79 <= L <= 93, we show that the
Lyapunov fractal dimension D scales microextensively, increasing linearly with
L even for increments Delta{L} that are small compared to the average cell size
of 9 and to various correlation lengths. This suggests that a spatially
homogeneous chaotic system does not have to increase its size by some
characteristic amount to increase its dynamical complexity, nor is the increase
in dimension related to the increase in the number of linearly unstable modes.Comment: 5 pages including 4 figures. Submitted to PR
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