REALTORUS

REALTORUS should be a dynamical system, which exhibits an invariant torus, either attracting or repelling. To design this system, we start in 2D:

$$\dot{r} A\cdot{}(r^2-R),\quad{}A\ne{}0,\ R>0$$ =

$$\dot{\rho}$$ = 1

Depending on the value of A this system has either an attracting or repelling cycle of radius R. In Cartesian coordinates ( $u=r\cdot{}\cos\rho,\ v=r\cdot{}\sin\rho$) this system is given as follows:

$$\dot{u} a\cdot{}u - v,\quad{}a=A\cdot{}(r-R/r),\ r=\sqrt{u^2+v^2}$$ =

$$\dot{v} a\cdot{}v + u$$ =

We now squeeze this system into half-plane  $\{(u',v') : u'>0\}$ by transformation  $u'=e^u,\ v'=v$ ( $u=\ln{}u',\ v=v'$) and end up with the following system:

$$\dot{u'}/u' a\cdot{}u - v,\quad{}a=A\cdot{}(r-R/r),\ r=\sqrt{u^2+v^2}$$ =

$$\dot{v'} a\cdot{}v + u$$ =

Using this dynamical system in 2D, we can construct a three-dimensional system, which actually contains an invariant torus by the following definition:

$$u'\cdot{}\cos\phi$$ x =

$$u'\cdot{}\sin\phi$$ y =

z = v'

Assuming $\dot{\phi}=C$ ( $u=\ln{}u'=\ln\sqrt{x^2+y^2},\ v=v'=z$) this dynamical system can be expressed as follows:

$$\dot{x} (a\cdot{}u-v)\cdot{}x - C\cdot{}y,\quad{}a\!=\!A\cdot{}(r-R/r),\ r\!=\!\sqrt{u^2+v^2}$$ =

$$\dot{y} (a\cdot{}u-v)\cdot{}y + C\cdot{}x$$ =

$$\dot{z} (a\cdot{}v+u)$$ =

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