A convection/reaction/switching system (original) (raw)

Modeling modal transitions in diffusing populations

2014

We consider a diffusing population as consisting of particles undergoing Brownian motion, each with its own history from a Lagrangian viewpoint. For each such particle we then consider possible ”state transitions” determined by crossing thresholds (hysteretic relay as in hybrid systems). Our objective here is to construct a continuum model of the resulting process as a reaction/diffusion system and then to show existence of ”solutions” of this system. Technical difficulties arise here in resolving the concerns of hybrid systems (anomalous points and the possibility of Zeno phenomena) in a setting where one is tracking the collective effects on individual diffusing particles without being able to track their individual trajectories. [For visualization, we think of an example of diffusing bacteria and nutrient in which, while undergoing diffusive motion, each bacterium is reacting to its own experience of local nutrient concentration in switching between dormant and active modes.] ∗E-...

Selection of periodic and localized states in Reaction-Diffusion-Advection systems

arXiv (Cornell University), 2008

Spatially periodic and localized states are theoretically studied in the context of reaction-diffusion-advection systems with mixed boundary conditions. The minimal requirement for emergence of nonuniform patterns is a two variable model with advection and only one diffusion term; this makes the pattern selection qualitatively different from the symmetric finite wave number Turing or Hopf instabilities. The key mechanism is coexistence of propagating or stationary nonuniform solutions, which are identified in a comoving reference. Consequently, stationary periodic states may stabilize if aperiodic boundary conditions are employed, since the translational symmetry which is essential for traveling waves, can not be preserved. In the same framework, we also explain propagation of traveling waves against the advective flow, and reveal the regime of excitable pulses.

Control design for suppressing transversal patterns in reaction–(convection)–diffusion systems

Journal of Process Control, 2006

The stabilization of planar stationary fronts solutions in a two-dimensional rectangular or cylinder domain, in which a diffusion-convection-reaction process occurs, is studied by reducing the original two-variable PDEs model to an approximate one-dimensional model that describes the behavior of the front line. We consider the control strategy based on sensors placed at the designed front line position and actuators that are spatially-uniform or space dependent. We present a systematic control design that determines the number of required sensors and actuators, their position and their form. The control used linear analysis of a lumped truncated model and concepts of finite and infinite zeros of linear multidimensional systems.

Hybrid switching diffusions: Continuity and differentiability

IEEE Conference on Decision and Control and European Control Conference, 2011

This work focuses on hybrid switching diffusion systems. After presenting the basic models, a difficult and fundamental problem, namely, well posedness is addressed. In particular, continuous and smooth dependence on initial data is treated. The main ideas are presented, whereas the verbatim proofs can be found in [35].

Drifting pattern domains in a reaction-diffusion system with nonlocal coupling

Physical review. E, Statistical, nonlinear, and soft matter physics, 2002

Drifting pattern domains (DPDs), i.e., moving localized patches of traveling waves embedded in a stationary (Turing) pattern background and vice versa, are observed in simulations of a reaction-diffusion model with nonlocal coupling. Within this model, a region of bistability between Turing patterns and traveling waves arises from a codimension-2 Turing-wave bifurcation (TWB). DPDs are found within that region in a substantial distance from the TWB. We investigated the dynamics of single interfaces between Turing and wave patterns. It is found that DPDs exist due to a locking of the interface velocities, which is imposed by the absence of space-time defects near these interfaces.

Bifurcation phenomena and cellular-pattern evolution in mixed-convection heat transfer

Journal of Fluid Mechanics, 1987

This investigation is concerned with the numerical calculation of multiple solutions for a mixed-convection flow problem in horizontal rectangular ducts. The numerical results are interpreted in terms of recent observations by Benjamin (1978a) on the bifurcation phenomena for a bounded incompressible fluid. The observed mutations of cellular flows are discussed in terms of dynamic interchange processes. Each cellular flow may be represented by a solution surface in the parametric space of Grashof number Gr and aspect ratio γ, which is delimited by stability boundaries. Such a stability map has been generated for each type of cellular flow by a series of numerical experiments. Once these boundaries are crossed one cellular flow mutates into another via a certain dynamical process. Although the nature of the singular points on this map have not been determined precisely, a plausible general structure of the cellular-flow exchange process emerges from this map with several features in ...

Spatio-temporal patterns in a reaction–diffusion system with wave instability

Chemical Engineering Science, 2000

We utilize a simple three-variable reaction}di!usion model to study patterns that emerge beyond the onset of the (short-)wave instability. We have found various wave patterns including standing waves, traveling waves, asymmetric standing}traveling waves and target patterns. We employ both periodic and zero #ux boundary conditions in the simulations, and we analyze the patterns using space}time two-dimensional Fourier spectra. A fascinating pattern of waves which periodically change their direction of propagation along a ring is found for very short systems. A related pattern of modulated standing waves is found for systems with zero #ux boundary conditions. In a two-dimensional system with small overcriticality we observe a wide variety of standing wave patterns. These include plain and modulated stripes, squares and rhombi. We also "nd standing waves consisting of periodic time sequences of stripes, rhombi and hexagons. The short-wave instability can lead to a much greater variety of spatio-temporal patterns than the aperiodic Turing and the long-wave oscillatory instabilities. For example, a single oscillatory cycle may display all the basic patterns related to the aperiodic Turing instability * stripes, hexagons and inverted hexagons (honeycomb) * as well as rhombi and modulated stripes. A rich plethora of patterns is seen in a system with cylindrical geometry * examples include rotating patterns of standing waves and counter-propagating waves.

Mode-locking in advection-reaction-diffusion systems: An invariant manifold perspective

Chaos (Woodbury, N.Y.), 2018

Fronts propagating in two-dimensional advection-reaction-diffusion systems exhibit a rich topological structure. When the underlying fluid flow is periodic in space and time, the reaction front can lock to the driving frequency. We explain this mode-locking phenomenon using the so-called burning invariant manifolds (BIMs). In fact, the mode-locked profile is delineated by a BIM attached to a relative periodic orbit (RPO) of the front element dynamics. Changes in the type (and loss) of mode-locking can be understood in terms of local and global bifurcations of the RPOs and their BIMs. We illustrate these concepts numerically using a chain of alternating vortices in a channel geometry.