Spatial bifurcation

Short description: Mathematical theory

Spatial bifurcation is a form of bifurcation theory. The classic bifurcation analysis is referred to as an ordinary differential equation system, which is independent on the spatial variables. However, most realistic systems are spatially dependent. In order to understand spatial variable system (partial differential equations), some scientists try to treat with the spatial variable as time and use the AUTO package[1] get a bifurcation results.[2][3]

The weak nonlinear analysis will not provide substantial insights into the nonlinear problem of pattern selection. To understand the pattern selection mechanism, the method of spatial dynamics is used,[4] which was found to be an effective method exploring the multiplicity of steady state solutions.[3][5]

## See also

* Spatial ecology
* spatial pattern

## References

1. ↑ "AUTO". http://indy.cs.concordia.ca/auto/.
2. ↑ Wang, R.H., Liu, Q.X., Sun, G.Q., Jin, Z., and Van de Koppel, J. (2010). "Nonlinear dynamic and pattern bifurcations in a model for spatial patterns in young mussel beds". Journal of the Royal Society Interface 6 (37): 705–18. doi:10.1098/rsif.2008.0439. PMID 18986965.
3. ↑ 3.0 3.1 A Yochelis (2008). "The formation of labyrinths, spots and stripe patterns in a biochemical approach to cardiovascular calcification". New J. Phys. 10 055002 (5): 055002. doi:10.1088/1367-2630/10/5/055002. Bibcode: 2008NJPh...10e5002Y.
4. ↑ Champneys A R (1998). "Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics". Physica D 112 (1–2): 158–86. doi:10.1016/S0167-2789(97)00209-1. Bibcode: 1998PhyD..112..158C.
5. ↑ Edgar Knobloch (2008). "Spatially localized structures in dissipative systems: open problems". Nonlinearity 21 (4): T45–60. doi:10.1088/0951-7715/21/4/T02. https://scholar.google.com/citations?view_op=view_citation&hl=en&user=4eI-QDkAAAAJ&citation_for_view=4eI-QDkAAAAJ:5nxA0vEk-isC.

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