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Spatially dependent parameter estimation and nonlinear data assimilation by autosynchronization of a system of partial differential equations
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10.1063/1.4812722
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1 Department of Mathematics, Clarkson University, Potsdam, New York 13669, USA
Chaos 23, 033101 (2013)
/content/aip/journal/chaos/23/3/10.1063/1.4812722
http://aip.metastore.ingenta.com/content/aip/journal/chaos/23/3/10.1063/1.4812722
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## Figures

FIG. 1.

Three sets of spatially dependent parameters used in simulations. Figures are described by Eq. , with on the left and on the right. Below, with the same ordering, are the parameters described by Eq. . Finally, the swirly parameters are shown in Figures .

FIG. 2.

Autosynchronization of species in Eqs. . Each figure shows drive (top) and response (bottom) pairs. (, 0) and in (a), (, 1000) and in (c), and (, 4788) and in (e). (, 0) and in (b), (, 1000) and in (d), and (, 4788) and in (f). Model parameters are and .

FIG. 3.

Autosynchronization of response parameters in Eqs. . Each figure shows drive (top) and response (bottom) pairs. and in (a), and in (c), and and in (e). and in (b), and in (d), and and in (f).

FIG. 4.

Autosynchronization of species in Eqs. . Each figure shows drive (top) and response (bottom) pairs. (, 0) and in (a), (, 1000) and in (c), and (, 10 660) and in (e). (, 0) and in (b), (, 1000) and in (d), and (, 10 660) and in (f). Model parameters are and .

FIG. 5.

Autosynchronization of response parameters in Eqs. . Each figure shows drive (top) and response (bottom) pairs. and in (a), and in (c), and and in (e). and in (b), and in (d), and and in (f).

FIG. 6.

Globally averaged relative synchronization error between drive and response PDE components and parameters on a log scale. Figures correspond to parameters built by Eq. and simulation displayed in Figures , respectively. Figures show globally averaged relative synchronization error for species and parameters built by Eq. , corresponding to simulations in Figures , respectively.

FIG. 7.

Autosynchronization of species in Eqs. . Each figure shows drive (top) and response (bottom) pairs. (, 0) and in (a), (, 1000) and in (c), and (, 9360) and in (e). (, 0) and in (b), (, 1000) and in (d), and (, 9360) and in (f). Model parameters are those shown in Figures .

FIG. 8.

Autosynchronization of parameters in Eqs. . Each figure shows drive (top) and response (bottom) pairs. and in (a), and in (c), and and in (e). and in (b), and in (d), and and in (f). Model parameters are those shown in Figures .

FIG. 9.

Globally averaged relative synchronization error between drive and response PDE components and parameters on a log scale, estimating perhaps more realistic spiral parameters. Figures (a) and (b) correspond to parameters shown in Figures and simulation displayed in Figures , respectively.

FIG. 10.

Locally averaged patches over which drive system is sampled shown in black. Sampled on subset of 3 × 3 grid points with a distance of 3grid points between patches.

FIG. 11.

Comparison of three different sampling schemes. Shown are relative synchronization errors between drive and response systems for sampling over 3 × 3 grid points (blue) with a distance of 3 grid points between subsequent patches, 2 × 2 grid points (red) with a distance of 2 grid points between subsequent patches, and 1 × 1 grid points (black) with a distance of 1 grid points between subsequent patches. Phytoplankton synchronization errors on left and zooplankton synchronization errors shown on right.

FIG. 12.

Autosynchronization results shown at t = 2000. Both species and both parameters shown compared with drive species and true parameters. Effect of adding diffusion to parameter equations is clearly visible in estimated parameters.

FIG. 13.

Globally averaged relative synchronization errors shown for species and parameters. Local sampling destroys stability of the identical synchronization manifold, however, spatial characteristics of parameters are still observed.

/content/aip/journal/chaos/23/3/10.1063/1.4812722
2013-07-08
2014-04-21

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