Vector field topology and local analysis

A common procedure to extract the topology of a dynamical system [6] is to proceed in the following way [5]:

1.

Identify all the critical points  c_i (also called fixed points, roots, etc.) by solving equation  f_p(c_i)=0.

2.

Investigate the Jacobian matrix of f_p at the critical points, i.e.,  $\left.\nabla\mathbf{f}_\mathbf{p}\right\vert _{\mathbf{c}_i}$. In the hyperbolic case this matrix of derivatives represents the major components of the flow near the critical points. Eigenvalues and eigenvectors intuitively describe the characteristics of the dynamics of $\mathbf{f}_\mathbf{p}(\mathbf{c}_i\!+\!\mathbf{d})$, i.e., of the flow near  c_i.

Depending on the local flow characteristics, critical points are classified as attractors, repellors, or saddles. Focal critical points exhibit a pair of conjugated complex eigenvalues, whereas nodes exhibit real eigenvalues only.

3.

Search for higher order characteristic elements, such as, e.g., cycles  $\mathcal{C}_i(t)$ ( $\mathcal{C}_i(t\!+\!T)=\mathcal{C}_i(t)$). Again perform an analysis of derivatives to learn about local flow characteristics.

4.

Extend the eigen-manifolds near the characteristic elements into phase space to determine their relation to other characteristic elements. Thereby structures such as separatrices, i.e., elements which divide the phase space into regions of qualitatively different dynamics, are identified.

Extracting location and shape of the topological elements, the geometry of behavior [1] is constructed. For a review of state of the art visualization techniques concerning flow topology see Chapter 2.

mailto:helwig@cg.tuwien.ac.at.