View source for Stability of equilibria

==Classification of Critical Cases==

Each critical case corresponds to a certain number, <math>k\ ,</math> of equality conditions (<math>\Phi=0</math>) imposed on the Taylor coefficients of <math>F(x)</math> at <math>x=c\ .</math> The number <math>k</math> is called the '''co-dimension'''. Cases of co-dimension <math>k</math> appear naturally when one considers families of dynamical systems <math>x'=F(x, s)</math> with <math>s=(s_1,\ldots,s_k)</math> (Arnold 1983). 
* Co-dimension 1 corresponds to the two simplest and the most important cases (Section [[#Two Major Critical Cases|Two Major Critical Cases]]): one zero eigenvalue and a pair of pure complex eigenvalues. 
* Co-dimension 2 includes the example in Section [[#More Complicated Example|More Complicated Example]].
* Co-dimension 3 includes the equilibrium in Section [[#More Complicated Example|More Complicated Example]], if, in addition, one of the following conditions is satisfied:
** The resonance condition <math>\omega_2=2\omega_1\ ,</math> or
** The condition <math>q_{11}=0\ .</math>
Notice that <math>q_{11}</math> is a quite complicated function of linear, quadratic, and cubic Taylor coefficients of <math>F(x)\ .</math> That is, the conditions are imposed not only on the Jacobian matrix, but also on the nonlinear part of the dynamical system.

All cases of co-dimension <math>k\leq3</math> are known for the general system \eqref{ode}. Arnold and Ilyashenko (1988) explain why <math>k=3</math> is a natural upper bound of the general theory. See also Khazin and Shnol (1991).

Systems with critical equilibria are not [[Structural Stability of Dynamical Systems|structurally stable]]: When parameters are perturbed, phase portraits of such systems undergo qualitative change - a [[bifurcation]] - near the equilibria. The first critical case (<math>\lambda=0</math>) corresponds to [[saddle-node bifurcation]]; the second case (<math>\lambda = \pm i \omega</math>) corresponds to [[Andronov-Hopf bifurcation]]; critical cases of co-dimensions 2 and 3 correspond to more complicated bifurcations, some of which are not well-understood.