function doesn't guarantee a limit cycle, torus, or chaos). Example: Let , , and . If then all eigenvalues are negative and the only attractor is a fixed point. If then the real part of one of the complex eigenvalue pair becomes positive and there is a strange attractor. The disappearance of this Lyapunov function coincides with a Hopf bifurcation. Line systems and eigenvalues It is also possible to arrange the species into a line. The interaction matrix for this system is very similar to that of a circle except the interaction terms in the lower left and upper right of the matrix are deleted (those that describe the interactions between species 1 and N, etc.). This change eliminates the Lyapunov function described above for the system on a circle, but most likely there are other Lyapunov functions that have not been discovered. The eigenvalues of the circle system plotted in the complex plane form a trefoil shape. The eigenvalues from a short line form a sideways Y, but those of a long line begin to resemble the trefoil shape of the circle. This could be due to the fact that a long line is indistinguishable from a circle to those species far from the ends. Notes Chaotic maps Equations Population ecology Community ecology Population models