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 and an attempt to derive this equation from hydrodynamic equations was done by Stuart for Plane Poiseuille flow in 1958. The formal derivation to derive the "Landau equation" was given by Stuart, Watson and Palm in 1960. The perturbation  in the vicinity of bifurcation is governed by the following equation where The evolution of the actual disturbance is given by the real part of formula_6 i.e., by formula_7. Here the real part of the growth rate is taken to be positive, i.e., formula_8 because otherwise the system is stable in the linear sense, that is to say, for infinitesimal disturbances (formula_9 is a small number), the nonlinear term in the above equation is negligible in comparison to the other two terms in which case the amplitude grows in time only if formula_8. The "Landau constant" is also taken to  be positive, formula_11 because otherwise the amplitude will grow indefinitely (see below equations and the general solution in the next section). The "Landau equation" is the equation for the magnitude of the disturbance, which can also be re-written as Similarly, the equation for the phase is given by Due to the universality of the equation, the equation finds its application in many fields such as hydrodynamic stability, Belousov–Zhabotinsky reaction, etc. General solution. The "Landau equation" is linear when it is written for the dependent variable formula_15, The general solution for formula_17 of the above equation is As formula_19 , the magnitude of the disturbance formula_9 approaches a constant value that is independent of its initial value, i.e., formula_21 when formula_22. The above solution implies that formula_9 does not have a real solution if formula_24 and formula_8. The associated solution for the phase function formula_26 is given by As formula_22, the phase varies linearly with time, formula_29