Editing Dynamical system (section)

===Conjugation results===
The results on the existence of a solution to the conjugation equation depend on the eigenvalues of ''J'' and the degree of smoothness required from ''h''.  As ''J'' does not need to have any special symmetries, its eigenvalues will typically be complex numbers.  When the eigenvalues of ''J'' are not in the unit circle, the dynamics near the fixed point ''x''<sub>0</sub> of ''F'' is called ''[[Hyperbolic fixed point|hyperbolic]]'' and when the eigenvalues are on the unit circle and complex, the dynamics is called ''elliptic''.

In the hyperbolic case, the [[Hartman–Grobman theorem]] gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map ''J''&nbsp;·&nbsp;''x''. The hyperbolic case is also ''structurally stable''.  Small changes in the vector field will only produce small changes in the Poincaré map and these small changes will reflect in small changes in the position of the eigenvalues of ''J'' in the complex plane, implying that the map is still hyperbolic.

The [[Kolmogorov–Arnold–Moser theorem|Kolmogorov–Arnold–Moser (KAM)]] theorem gives the behavior near an elliptic point.