c,
where Re is the Reynolds number and Rec is the critical Reynolds number.
Flow that is not laminar is said to be turbulent.
Faraday waves:
Surface waves on a liquid air interface excited by a vertical vibration of a fluid layer.
A sufficiently large excitation amplitude the plane interface undergoes an instability (Faraday instability)
and standing surface waves appear, oscillating with a frequency one half of the drive.
This type of parametric wave instability is attractive as the wavelength of the pattern is dispersion
rather than geometry controlled.
Just by varying the drive frequency the wave number can be tuned in a wide range.
In that sense the Faraday setup is well suited for the study of phase dynamics.
-
One approach is the amplitude equation is based on the linear instability of a homogeneous state and leads naturally to a classification of patterns in terms of characteristic wave numbers and frequencies.
-
A different but equally universal description, the phase dynamics
applies to situations where a periodic spatial pattern experiences long wavelength phase modulations.
Amplitude equations:
describe the slow spatial and temporal variation of a system in the vicinity of a linear instability.
A prototypical amplitude equation is the complex Ginzburg-Landau equation
(CGLE)
which describes the behavior of extended systems near a Hopf bifurcation of a steady, homogeneous state to a limit cycle.
In particular, the CGLE applies to
reaction-diffusion systems.
This approach is helpful for studying the characteristic generic aspects of the Hopf bifurcation because all systems behave in a similar manner sufficiently close to the onset of oscillations as a consequence of the center manifold theorem. On the other hand, the validity of this approach is limited to the vicinity of the bifurcation point.
Active/passive modes :
Modes with positive(negative) real part eigenvalues from linear stability analysis are active(passive) modes.
The active modes are those responsible for the loss of stability of the reference state by growing exponentially.
On the other hand, there exist so-called passive modes, the linear frequencies of which are still damped above the critical point.
They however also come into the determination of pattern as they are continuously regenerated by the nonlinear interactions between members of the active set.
Their dynamics results from the balance between this regeneration and their rapid linear decay.
[P. Borckmans, G. Dewel, A. DeWit, and D.Walgraef, in Chemical Waves and Patterns, edited by R. Kapral and K. Showalter (Kluwer, Dordrecht, 1995), p. 323.]
Reaction-diffusion systems:
Mathematically a reaction-diffusion system is obtained by adding some diffusion terms to a set of ordinary differential equations (ODE) which are first order in time.
The reaction-diffusion model is literally an appreciated model for studying the dynamics of chemically reacting and diffusing systems.
Actually, the scope of this model is much wider. For instance, in the field of biology,
the propagation of the action potential in nerves and nervelike tissues is known to obey this type of equation,
and some mathematical ecologists employ reaction-diffusion models for explaining various ecological patterns observed in nature.
The total system can be viewed as an assembly of a large number of identical local systems which are diffusion-coupled to each other.
[from Kuramoto book 1984]
Stationary and oscillatory bifurcations:
Two typical distributions of the eigenvalues at the criticality
a) one eigenvalue on the real axis crosses the origin (im=0 means stationary, for example, Turing),
(b) a pair of complex-conjugate eigenvalues cross the imaginary axis simultaneously (Hopf).
Turing instability/Morphological instability/Diffusion driven instability: (click to see) 
that satisfy 
where A is an 
matrix and x is some vector. In this
case, x is called an eigenvector.
that satisfy 
where 
is an 
periodic
matrix and 
is a periodic vector.
,
where 
are 
for which there are infinately many periodic
solutions passing through any region in phase space.)
then the
linearization is the dynamical system, 
where A is the
derivative of F evaluated at a solution.
, the period,
comes back to its initial point.
.
.
(
It is described by the Hamiltonian
i.e.
.
is a nondimensionalized version of the equation 
derived by Korteweg and de Vries (1895) which described weakly nonlinear shallow water waves.
h is the channel height,
T is the surface tension,
g is the gravitational acceleration,
and $\rho$ is the density.
This equation was found to have solitary wave solutions, vindicating the observations made 51 years earlier of a solitary channel wave by Russell (1844).
then exists.
If the inner product vanishes we say
the two vectors are orthogonal.
: 
is the matrix obtained by exchanging A's rows and columns.
(or 
) is a matrix obtained from a given matrix by taking
the complex conjugate of each element.
where 
and where 
is define by 
=
: 
=adj A, is formed by replacing each entry of A be its 
: 
.
:
times the determinant of the matrix obtained by deleting the i-th row and j-th column of A.
(i.e., the vectors are perpendicular) are said to be orthogonal.
that satisfy 
.
diffusion tensors, the 
are nonlinear evaluation functions,
while 
represents the external constraints or bifurcation parameters.
The first instabilities or primary bifurcations are determined by the linear
stability analysis around the equilibrium state 
.
In the isotropic case:
