The parameter map of a partial differen

Tim Hutton - 2014-06-30 14:04:00+0000 - Updated: 2014-07-01 08:06:58+0000

The parameter map of a partial differential equation, showing a reasonably standard pattern of Turing instabilities for different combinations of parameters - spots and labyrinthine stripes. But..

The intriguing thing here is that this is a single chemical system! It doesn't fit into the usual characterisation of activator and inhibitor, like Gray-Scott or Turing's original model.

The formula is:

du/dt = mu*u - bilaplacian(u) - 2*laplacian(u) - u + beta*u*u - u*u*u

In this parameter map, mu ranges from -1 on the left to 5 on the right, and beta ranges from -3 at the bottom to 3 at the top.

The bilaplacian is the laplacian operator applied twice. In a one-dimensional system if your laplacian is computed by a finite differencing kernel of 1,-2,1 then the bilaplacian kernel is 1,-4,6,-4,1.

The formula comes from the work of +Matt Pennybacker:
http://pennybacker.net
(Matt suggested trying it without the gradient term in the paper.)

There are some Ready files here to explore:
http://code.google.com/p/reaction-diffusion/source/browse/trunk/Ready#Ready%2FPatterns%2FPennybacker2013

The parameter map of a partial differential equation, showing a reasonably standard pattern of Turing instabilities for different combinations of parameters - spots and labyrinthine stripes. The intriguing thing here is that this is a single chemical system! It doesn't fit into the usual characterisation of activator and inhibitor, like Gray-Scott or Turing's original model. The formula is: du/dt = mu*u - bilaplacian(u) - 2*laplacian(u) - u + beta*u*u - u*u*u In this parameter map, mu ranges from -1 on the left to 1 on the right, and beta ranges from -3 at the bottom to 3 at the top. The bilaplacian is the laplacian operator applied twice. In a one-dimensional system if your laplacian is computed by a finite differencing kernel of 1,-2,1 then the bilaplacian kernel is 1,-4,6,-4,1. The formula comes from the work of +Matt Pennybacker: http://pennybacker.net (Matt suggested trying it without the gradient term in the paper.) There are some Ready files here to explore: http://code.google.com/p/reaction-diffusion/source/browse/trunk/Ready#Ready%2FPatterns%2FPennybacker2013

Shared with: Public, Matt Pennybacker

+1'd by: Martin Novy, Abdi Osman, Torolf Sauermann, Aistis Raulinaitis, Cassidy Curtis, Aaron Hertzmann, Craig Reynolds, Elżbieta Zadora-Zuzelska, Luc Blanchette, Ricky Moore-Daniels, Paweł Zuzelski, Ángel Linares García, Felix Woitzel, Štěpán Roučka, Hector Yee, Kevin O'Bryant, Hunter Yavitz, Sebastian Fernandez Alberdi, Dan Wills, Dylan Leigh, Dave Gordon, Kevin C., kenny bell, Daniel Ammann, Owen Maresh, Doug Hackworth, allan de medeiros martins, Emil Mikulic, Scott Vorthmann

Reshared by: Martin Novy, Evelyn Mitchell, Moshe Vardi, Lee Nelson, Alberto Pereira, Paweł Zuzelski, Felix Woitzel, Dave Gordon, Mirko Bulaja, Stephen Loftus-Mercer

Dan Wills - 2014-07-01 01:51:39+0000

Wow Tim that is bloody interesting! Thanks for sharing, both to yourself and Matt Pennybacker!

Christopher Hanusa - 2014-07-01 02:27:14+0000

This makes me think about a random grove: http://arxiv.org/abs/math/0407171
(See page 10).  I wonder if it is at all related.

Felix Woitzel - 2014-07-01 06:46:55+0000

it's really reminiscent of http://mrob.com/pub/comp/xmorphia/index.html - good job! 

Lewpen Kinross-Skeels - 2014-07-16 20:56:58+0000

Beautiful!

This post was originally on Google+