Stuart–Landau equation

The Stuart–Landau equation describes the behavior of a nonlinear oscillating system near the Hopf bifurcation, named after John Trevor Stuart and Lev Landau. In 1944, Landau proposed an equation for the evolution of the magnitude of the disturbance, which is now called as the Landau equation, to explain the transition to turbulence without providing a formal derivation[1] and an attempt to derive this equation from hydrodynamic equations was done by Stuart for Plane Poiseuille flow in 1958.[2] The formal derivation to derive Landau equation was given by Stuart, Watson and Palm in 1960.[3][4][5] The perturbation in the vicinity of bifurcation is governed by the following equation

${\displaystyle {\frac {dA}{dt}}=\sigma A-{\frac {l}{2}}A|A|^{2}.}$

where

• ${\displaystyle A=|A|e^{i\phi }}$ is a complex quantity describing the disturbance,
• ${\displaystyle \sigma =\sigma _{r}+i\sigma _{i}}$ is the complex growth rate,
• ${\displaystyle l=l_{r}+il_{i}}$ is a complex number and ${\displaystyle l_{r}}$ is the Landau constant .

The Landau equation is the equation for the magnitude of the disturbance

${\displaystyle {\frac {d|A|^{2}}{dt}}=2\sigma _{r}|A|^{2}-l_{r}|A|^{4},}$

can also be re-written as[6]

${\displaystyle {\frac {d|A|}{dt}}=\sigma _{r}|A|-{\frac {l_{r}}{2}}|A|^{3}.}$

Similarly, the equation for the phase is given by

${\displaystyle {\frac {d\phi }{dt}}=\sigma _{i}-{\frac {l_{i}}{2}}|A|^{2}.}$

Due to the universality of the equation, the equation finds its application in many fields such as hydrodynamic stability,[7][8] chemical reactions[9] such as Belousov–Zhabotinsky reaction, etc.

The Landau equation is linear when written for the dependent variable ${\displaystyle |A|^{-2}}$,

${\displaystyle {\frac {d|A|^{-2}}{dt}}+2\sigma _{r}|A|^{-2}=l_{r}}$

leading to the general solution (for ${\displaystyle \sigma _{r}\neq 0}$)

${\displaystyle |A|^{-2}={\frac {l_{r}}{2\sigma _{r}}}+\left(A_{o}^{-2}-{\frac {l_{r}}{2\sigma _{r}}}\right)e^{-2\sigma _{r}t}}$

where ${\displaystyle A_{o}=A(0)}$. As ${\displaystyle t\rightarrow \infty }$, the solution goes to a constant value, independent of the initial condition,, i.e., ${\displaystyle |A|\rightarrow (2\sigma _{r}/l_{r})^{1/2}}$ at large times.