Stuart-Landau equation

US to calculate explicitly a certain constant (called the Landau constant) appearing in the Stuart-Landau equation, whose sign determines the stability of the bifurcating periodic solution. Otherwise, the Stuart-Landau equation itself is not likely to arouse much theoretical interest, although it may have some value in serving as an ideal nonlinear oscillator model. 

Then the solvability condition (2.2.16) itself turns out to take the form of the Stuart-Landau equation... 

When D deviates slightly from ) one may still expect that the dominant part of Qiiff, t) preserves the form of (5.7.4) except that IF should then be interpreted as a small-amplitude and slowly varying function of t. As shown below, the application of the theory of Sect. 2.2 reveals that IF obeys the Stuart-Landau equation near criticality. 

From (9.21a), with (9.22), the fundamental harmonic Ai obeys the following Stuart - Landau type equation ... 

The derivation of the amplitude, Stuart - Landau type, equation near criticality using the technique of multiple scales is now well known and the details can be found in Newell (1974) or Stewartson and Stuart (1971). 

We perform a nonlinear analysis that will allow us to obtain amplitude equations to characterize the evolution of the unstable modes. Gross and Volpert (4) have studied the 1-D case for loss of neutral stability at the wavenumber s = 0. This corresponds to sufficiently small values of the tube circumference L. The analysis in this instance results in a single Landau-Stuart equation which governs the weakly unstable modes. If, however, the tube circumference L is large, loss of stability will occur for some s > 0. Our nonlinear analysis yields a coupled set of Landau amplitude equations. 

From IBL Averaged Shkadov System to Landau - Stuart Evolution NL Equation... 

But for a weakly nonlinear case, if we want to derive the associated Landau -Stuart (LS envelope) evolution equation, it is suffisant to assume that the amplitude of the wave packet envelope is a function of 2 = 0 only. 

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