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Specific external forcing of spatiotemporal dynamics in reaction–diffusion systems
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10.1063/1.1886285
/content/aip/journal/chaos/15/2/10.1063/1.1886285
http://aip.metastore.ingenta.com/content/aip/journal/chaos/15/2/10.1063/1.1886285
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Figures

Image of FIG. 1.
FIG. 1.

Spatially distributed influx control scenario for a quasi-1D system modeled by equation system (1) . In a finite volume discretization of the PDE system the additional control influx into the finite volume cells is modeled in a linear approach. is the unity vector pointing in outward direction of the cell and the corresponding directional derivative. is the concentration of chemoattractant in the external reservoir connected to the finite volume cell , is chosen for all control scenarios (see Figs. 2–4 ) with the spatial domain , a spatial grid step size of 0.005 [confirmed to give accurate results in numerical simulations of the chemotaxis system (Refs. 30 and 31 )] and an external reservoir connected to every tenth finite volume cell.

Image of FIG. 2.
FIG. 2.

Induction and stabilization of a spatial cell distribution pattern . Cell densities for three different time points and exemplarily some corresponding control functions , in the external reservoirs are shown. are assumed to be piecewise constant on small multiple shooting time intervals . Integrator accuracy, , KKT (Karush–Kuhn–Tucker) tolerance (measuring the deviation from necessary condition for optimal solution), , 29 SQP iterations; total computing time, , approximately 80% needed for computation of numerical derivatives and 20% for the SQP iterations.

Image of FIG. 3.
FIG. 3.

Induction and stabilization of a spatial cell distribution pattern . Cell densities for three different time points and a selection of corresponding control functions , in the external reservoirs are shown. are assumed to be piecewise linear on small multiple shooting time intervals . Integrator accuracy, , KKT (Karush–Kuhn–Tucker) tolerance (measuring the deviation from necessary condition for optimal solution), , 26 SQP iterations; total computing time, , approximately 80% needed for computation of numerical derivatives and 20% for the SQP iterations.

Image of FIG. 4.
FIG. 4.

Induction and tracking of a propagating wave , propagation velocity . Cell densities wave profiles for four different time points and a selection of corresponding control functions , in the external reservoirs are shown. are assumed to be piecewise constant on small multiple shooting time intervals . Integrator accuracy, , KKT (Karush–Kuhn–Tucker) tolerance (measuring the deviation from necessary condition for optimal solution), , 47 SQP iterations; total computing time, , approximately 80% needed for computation of numerical derivatives and 20% for the SQP iterations.

Image of FIG. 5.
FIG. 5.

Multiple shooting discretization and decoupled piecewise trajectory integration for given initial values at the multiple shooting nodes and control function parameters (here, piecewise constant).

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/content/aip/journal/chaos/15/2/10.1063/1.1886285
2005-04-07
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Specific external forcing of spatiotemporal dynamics in reaction–diffusion systems
http://aip.metastore.ingenta.com/content/aip/journal/chaos/15/2/10.1063/1.1886285
10.1063/1.1886285
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