Landau Equation and Multiple Hopf-Bifurcation

Abstract

The linear theory of stability of a steady basic flow generally gives a spectrum of independent modes with velocity perturbation of the form,

$$ u'(\overrightarrow X ,t) = \sum\limits_{j = 1}^\infty {Aj(t)fj} (\overrightarrow X ) + A_j^* (t)f_j^* (\overrightarrow X ) $$

((5.1.1))

where the quantities with asterisks denote complex conjugate. In the linear stability theory we generally focus upon one mode at a time- the so-called normal mode analysis. If the complex amplitude of any one of the mode that grows with time is given by

$$ Aj(t) = Const.e^{s_j t} $$

((5.1.2))

then it is easy to see that the evolution equation for the amplitude of this mode is given by,

$$ \frac{{dAj}} {{dt}} = sjAj $$

((5.1.3))