Square Turing patterns

The cubic amplitude equations:

To split the real and imaginary parts, let . Modulus equation can be obtained.

It has three types of stationary solutions. Their stabilities are determined by eigenvalues of the Jacobian matrix

Solution	Eigenvalues	Stable conditions
I. Trivial uniform solution
II. One stripe solution		,
III. One square lattice		,

An ovweview:

Actually, here "stable" does not mean definitely the system will approch. Higher symmetric (hex) patterns should be checked.

mu=-0.5,g1=1,g2=0.5,h1=0.32,h2=0.2,h12=0

The quintic amplitude equation for square lattices

It has 4 types of solutions:

1. A trivial one
2. Two square solutions are

   The limit-point sits:

   Eigenvalues are:

   Stable squares require two negative eigenvalues. The first eigenvalue is not a problem (it starts from 0 at the limit point, declines monotonically), problem is the 2nd one. At the limit-point, .

   Born stable happens when , otherwise, it is born unstable until mu reaches .

   This switching occurs at mu2=0 gives

   1. When , (g1<0) the bif. turns into a subcritical one. The limit-point reaches mu1. The top branch is always stable, the other is a saddle.
   2. When both are positive, the bif. is, of curse, a subcritical one. For example, g1=1,g2=0.5;h2=0.2;h12=0.
      h2=0 --- unstable always --- 0.2 --- stable after mu2 --- 0.4 --- always stable.
3. Two stripes:

   The limit-point sits:

   Eigenvalues are:

   Subcritical case is born stable always of positive g2 and h2 as our focus. It turns unstable until .

   The switching occurs at mu2=0 gives .

4. Two mixtures:

   Their stabilities are determined by eigenvalues of the Jacobian matrix:
   

To hexagonal Turing