1 / 36

Spatially-extended dynamical systems and pattern formation Torino, January-March 2010

Spatially-extended dynamical systems and pattern formation Torino, January-March 2010 a.provenzale@isac.cnr.it. Goal: provide an introduction to the behavior of spatially-extended dynamical systems and to the study of PDEs. an introduction to pattern formation

dextra
Download Presentation

Spatially-extended dynamical systems and pattern formation Torino, January-March 2010

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Spatially-extended dynamical systems and pattern formation Torino, January-March 2010 a.provenzale@isac.cnr.it

  2. Goal: provide an introduction to the behavior of spatially-extended dynamical systems and to the study of PDEs. an introduction to pattern formation and the study of nonlinear PDEs applications: geosciences, biology, ecology

  3. Pattern: An organized, long-lived non-homogeneous state of a physical, chemical or biological system.

  4. Physical patterns Convection: S. Ciliberto et al., Phys. Rev. Lett. 61, 1198 (1988)

  5. Physical patterns Atmospheric convection. Photo by Hezi Yizhaq, Sede Boker, Negev desert

  6. Patterns Geomorphology: aeolian ripples. Wadi Rum desert, Jordan Geomorphology: aeolian ripples. Wadi Rum desert

  7. Erosion patterns in badlands (Tzin valley, Israel)

  8. http://daac.gsfc.nasa.gov/ DAAC_DOCS/geomorphology River networks (Yemen)

  9. Chemical and biological patterns A.T. Winfree et al,  Phisica D8, 35 (1983)

  10. Biological patterns S. Kondo et al, Nature 376, 765 (1995) J.D. Murray, J. Theor. Biol. 88, 161 (1981)

  11. Vegetation patterns at landscape scale 50 m Rietkerk et al., The American Naturalist 160 (4), 2002

  12. Vegetation patterns at landscape scale Valentin et al., Catena 37, 1-24 (1999)

  13. Coherent structures: vortices

  14. Coherent structures: vortices

  15. Coherent structures: vortices

  16. Coherent structures: vortices

  17. Coherent structures: convective plumes

  18. Contents: • 1. Intro to extended systems and PDE: heat eq., Fisher eq. • 2. Navier-Stokes eqns. and fluid dynamics • 3. Convection: linear stability, nonlinear saturation, patterns • 4. Turbulence and transport • 5. The Atmospheric Planetary Boundary Layer • 6. Rotating fluids, waves and vortices • 7. Turing mechanism and chemical patterns • 8. Patterns in geomorphology: aeolian ripples • 9. Vegetation patterns

  19. Introduction to spatially-extended dynamical systems Types of mathematical descriptions: PDEs (t=c, x=c, f=c) Coupled ODEs (t=c, x=d, f=c) Coupled maps (t=d, x=d, f=c) Cellular automata (t=d, x=d, f=d)

  20. PDE for time evolution: initial value problem

  21. PDE for boundary value problems

  22. PDE for time evolution: look for stationary solutions, homogeneous solutions, special solutions (eg travelling waves), general solutions

  23. The simplest PDE

  24. Characteristics t x

  25. Another approach: Fourier decomposition

  26. Another approach: Fourier decomposition

  27. This is a linear equation (superposition) This is a non-dispersive equation This is a conservative equation

  28. This is a conservative equation (also, the amplitude of each Fourier component)

  29. A slightly more complicated equation Still linear and conservative, but now dispersive

  30. A different case Still linear but now dissipative !

  31. Homogeneous state and its stability

  32. Finally, our most complicated linear PDE…

More Related