COMPLFP

In addition to the case that all eigenvalues/-vectors of the critical point's Jacobian matrix are real, a pair of conjugated complex eigenvalues/-vectors can occur. This case is not possible in dynamical system REALFP, thus another system is necessary to test this case - we start with r and $\phi$ denoted in polar coordinates:

$$\dot{r} A\cdot{}r,\quad{}A\ne{}0$$ =

$$\dot{\phi}$$ = 1

$$\dot{z} B\cdot{}z$$ =

Depending on the values of A and B this system exhibits either an attracting, repelling, or saddle critical point with a pair of conjugated complex eigenvalues/-vectors:

$$A<0,\ B<0 \to$$ attracting focus

$$A<0,\ B>0 \to$$ saddle focus (attracting z axis)

$$A>0,\ B<0 \to$$ saddle focus (repelling z axis)

$$A>0,\ B>0 \to$$ repelling focus

In terms of Cartesian coordinates ( $x=r\cdot{}\cos\phi,\ y=r\cdot{}\sin\phi$) this dynamical system can be written as

$$\dot{x} A\cdot{}x - y$$ =

$$\dot{y} A\cdot{}y + x$$ =

$$\dot{z} B\cdot{}z$$ =

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