Let 
be an equilibrium point of a vector field.
Then 
is called a hyperbolic
fixed point if none of the real parts of the eigenvalues of 
is equal to zero. The test for asymptotic stability of the
previous section can be restated as: a hyperbolic fixed point is stable if the
real parts of all its eigenvalues are negative.
A fixed point of a map is hyperbolic if none of the moduli of the eigenvalues
equals one.
The motion near a hyperbolic fixed point can be analyzed and
brought into a standard form by a linear transformation to the
eigenvectors of . Additional analysis,
including higher-order terms, is usually needed to analyze the
motion
near a nonhyperbolic fixed point.
At last, we can precisely define the terms saddle, sink, source, and center. A hyperbolic equilibrium solution is a saddle if the real part of at least one eigenvalue of the linearized vector field is less than zero and if the real part of at least one eigenvalue is greater than zero. Similarly, a saddle point for a map is a hyperbolic point if at least one of the eigenvalues of the associated linear map has a modulus greater than one, and if one of the eigenvalues has modulus less than one.
A hyperbolic point of a flow is a stable node or sink if all the eigenvalues have real parts less than zero. Similarly, if all the moduli are less than one then the hyperbolic point of a map is a sink.
A hyperbolic point is an unstable node or source if the real parts of all the eigenvalues are greater than zero. The moduli of a source of a map are all greater than one.
A center is a nonhyperbolic fixed point for which all the eigenvalues are purely imaginary and nonzero (modulus one for maps). For a picture of the elementary equilibrium points in three-dimensional space see Figure 3.10 of Thompson and Stewart [7]. The corresponding stability information for a hyperbolic fixed point of a two-dimensional map is summarized in Figure 4.9.
Figure 4.9: Complex eigenvalues for a two-dimensional map with a hyperbolic
fixed point: (a) saddle, (b)
sink, and (c) source.