29,579 research outputs found
Finite-parameter feedback control for stabilizing the complex Ginzburg-Landau equation
In this paper, we prove the exponential stabilization of solutions for
complex Ginzburg-Landau equations using finite-parameter feedback control
algorithms, which employ finitely many volume elements, Fourier modes or nodal
observables (controllers). We also propose a feedback control for steering
solutions of the Ginzburg-Landau equation to a desired solution of the
non-controlled system. In this latter problem, the feedback controller also
involves the measurement of the solution to the non-controlled system.Comment: 20 page
Stationary localized structures and the effect of the delayed feedback in the Brusselator model
The Brusselator reaction-diffusion model is a paradigm for the understanding
of dissipative structures in systems out of equilibrium. In the first part of
this paper, we investigate the formation of stationary localized structures in
the Brusselator model. By using numerical continuation methods in two spatial
dimensions, we establish a bifurcation diagram showing the emergence of
localized spots. We characterize the transition from a single spot to an
extended pattern in the form of squares. In the second part, we incorporate
delayed feedback control and show that delayed feedback can induce a
spontaneous motion of both localized and periodic dissipative structures. We
characterize this motion by estimating the threshold and the velocity of the
moving dissipative structures.Comment: 18 pages, 11 figure
Partial differential equations for self-organization in cellular and developmental biology
Understanding the mechanisms governing and regulating the emergence of structure and heterogeneity within cellular systems, such as the developing embryo, represents a multiscale challenge typifying current integrative biology research, namely, explaining the macroscale behaviour of a system from microscale dynamics. This review will focus upon modelling how cell-based dynamics orchestrate the emergence of higher level structure. After surveying representative biological examples and the models used to describe them, we will assess how developments at the scale of molecular biology have impacted on current theoretical frameworks, and the new modelling opportunities that are emerging as a result. We shall restrict our survey of mathematical approaches to partial differential equations and the tools required for their analysis. We will discuss the gap between the modelling abstraction and biological reality, the challenges this presents and highlight some open problems in the field
Brownian Molecules Formed by Delayed Harmonic Interactions
A time-delayed response of individual living organisms to information
exchanged within flocks or swarms leads to the emergence of complex collective
behaviors. A recent experimental setup by (Khadka et al 2018 Nat. Commun. 9
3864), employing synthetic microswimmers, allows to emulate and study such
behavior in a controlled way, in the lab. Motivated by these experiments, we
study a system of N Brownian particles interacting via a retarded harmonic
interaction. For , we characterize its collective behavior
analytically, by solving the pertinent stochastic delay-differential equations,
and for by Brownian dynamics simulations. The particles form
molecule-like non-equilibrium structures which become unstable with increasing
number of particles, delay time, and interaction strength. We evaluate the
entropy and information fluxes maintaining these structures and, to
quantitatively characterize their stability, develop an approximate
time-dependent transition-state theory to characterize transitions between
different isomers of the molecules. For completeness, we include a
comprehensive discussion of the analytical solution procedure for systems of
linear stochastic delay differential equations in finite dimension, and new
results for covariance and time-correlation matrices.Comment: 36 pages, 26 figures, current version: further improvements and one
correctio
Delay induced Turing-like waves for one species reaction-diffusion model on a network
A one species time-delay reaction-diffusion system defined on a complex
networks is studied. Travelling waves are predicted to occur as follows a
symmetry breaking instability of an homogenous stationary stable solution,
subject to an external non homogenous perturbation. These are generalized
Turing-like waves that materialize in a single species populations dynamics
model, as the unexpected byproduct of the imposed delay in the diffusion part.
Sufficient conditions for the onset of the instability are mathematically
provided by performing a linear stability analysis adapted to time delayed
differential equation. The method here developed exploits the properties of the
Lambert W-function. The prediction of the theory are confirmed by direct
numerical simulation carried out for a modified version of the classical Fisher
model, defined on a Watts-Strogatz networks and with the inclusion of the
delay
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