The parameter map of a partial differential equation, showing a reasonably st...
- 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 :
http://pennybacker.net
(Matt suggested trying it without the gradient term in the paper.)

There are some Ready files here to explore:

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
- 2014-07-01 01:51:39+0000
Wow Tim that is bloody interesting! Thanks for sharing, both to yourself and Matt Pennybacker!
- 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.
- 2014-07-01 06:46:55+0000
it's really reminiscent of http://mrob.com/pub/comp/xmorphia/index.html - good job!
- 2014-07-16 20:56:58+0000
Beautiful!

This post was originally on Google+