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Stuart-Landau Equation

The Stuart-Landau equation is a cubic ordinary differential equation for the complex amplitude of an oscillation near a Hopf bifurcation. In a common normalization it is

z^'(t)=(mu+iomega)z(t)-(a+ib)|z(t)|^2z(t),
(1)

where primes denote derivatives with respect to time t and mu, omega, a, and b are real constants. The equation arose in weakly nonlinear analyses of fluid-flow instability (Stuart 1960, Watson 1960).

In polar coordinates z(t)=r(t)e^(itheta(t)) with r(t)>0, the amplitude and phase satisfy

r^'(t)=mur(t)-ar(t)^3
(2)
theta^'(t)=omega-br(t)^2.
(3)

If a>0 and mu>0, the nonzero equilibrium radius is r_*=sqrt(mu/a). The derivative of the radial right-hand side at r_* is -2mu, so nearby radii approach r_*. If the angular speed omega-bmu/a is nonzero, the circle of radius r_* is an attracting limit cycle. When omega!=0 and a>0, this cycle emerges from the origin as mu passes through zero, giving a supercritical Hopf bifurcation (Guckenheimer and Holmes 1983, pp. 150-154).

See also

Hopf Bifurcation, Limit Cycle, Ordinary Differential Equation

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References

Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag, pp. 150-154, 1983. https://doi.org/10.1007/978-1-4612-1140-2.Stuart, J. T. "On the Non-Linear Mechanics of Wave Disturbances in Stable and Unstable Parallel Flows. Part 1. The Basic Behaviour in Plane Poiseuille Flow." J. Fluid Mech. 9, 353-370, 1960. https://doi.org/10.1017/S002211206000116X.Watson, J. "On the Non-Linear Mechanics of Wave Disturbances in Stable and Unstable Parallel Flows. Part 2. The Development of a Solution for Plane Poiseuille Flow and for Plane Couette Flow." J. Fluid Mech. 9, 371-389, 1960. https://doi.org/10.1017/S0022112060001171.

Cite this as:

Weisstein, Eric W. "Stuart-Landau Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Stuart-LandauEquation.html

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