equation for stops the growth in . Fixed points In order to find the fixed points, the three Rössler equations are set to zero and the (,,) coordinates of each fixed point were determined by solving the resulting equations. This yields the general equations of each of the fixed point coordinates: Which in turn can be used to show the actual fixed points for a given set of parameter values: As shown in the general plots of the Rössler Attractor above, one of these fixed points resides in the center of the attractor loop and the other lies relatively far from the attractor. Eigenvalues and eigenvectors The stability of each of these fixed points can be analyzed by determining their respective eigenvalues and eigenvectors. Beginning with the Jacobian: the eigenvalues can be determined by solving the following cubic: For the centrally located fixed point, Rössler's original parameter values of a=0.2, b=0.2, and c=5.7 yield eigenvalues of: The magnitude of a negative eigenvalue characterizes the level of attraction along the corresponding eigenvector. Similarly the magnitude of a positive eigenvalue characterizes the level of repulsion along the corresponding eigenvector. The eigenvectors corresponding to these eigenvalues are: These eigenvectors have several interesting implications. First, the two eigenvalue/eigenvector pairs ( and ) are responsible for the steady outward slide that occurs in the main disk of the attractor. The last eigenvalue/eigenvector pair is attracting along an axis that runs through the center of the manifold and accounts for the z motion that occurs within the attractor. This effect is roughly demonstrated with the figure below. The figure examines the central fixed point eigenvectors. The blue line corresponds to the standard Rössler attractor generated with , , and . The red dot in the center of this attractor is . The red line intersecting that fixed point is an illustration of the repulsing plane generated by and . The green line is an illustration of the attracting . The magenta line is generated by stepping backwards through time from a point on the attracting eigenvector which is slightly above – it illustrates the behavior of points that become completely dominated by that vector. Note that the magenta line nearly touches the plane of the attractor before being pulled upwards into the fixed point; this suggests that the general appearance and behavior of the Rössler attractor is largely a product of the interaction between the attracting