REALCYC

For testing Poincaré maps we define a simple cycle in 3D. The critical point of the Poincaré map of this dynamical system should be either an attracting node, a saddle, or a repelling node. For this purpose we think about points in phase space as specified by coordinates r, $\phi$, and z: r and $\phi$ denote polar coordinates in the x-y plane. REALCYC can now be specified as follows:

$$\dot{r} A\cdot{}(r^2-1),\quad{}A\ne{}0$$ =

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

$$\dot{z} B\cdot{}z,\quad{}B\ne{}0$$ =

This dynamical system contains a cycle. This is easy to show: $\dot{r}\vert _{r=1}=0$ and $\dot{z}\vert _{z=0}=0$ induce that the unit cycle $\mathcal{C}=\{\mathbf{x}:r=1\}$ is a cycle of this dynamical system. Furthermore it can be stated that there are no other cycles than $\mathcal{C}$ in this specific dynamical system: since $\dot{z}\vert _{z\ne{}0}\ne{}0$ no cycle can exist outside the x-y plane; $\dot{r}\vert _{\vert r\vert\ne{}1}\ne0$ assures that no other cycle than the unit cycle resides within the x-y plane.

Using this definition, we can directly influence the Poincaré map of cycle  $\mathcal{C}$ by adjusting parameters A and B:

$$A<0,\ B<0 \to \textrm{stable limit cycle } \mathcal{C}$$

$$A<0,\ B>0 \to \textrm{saddle cycle } \mathcal{C}, \textrm{ attracting cylinder } r=1$$

$$A>0,\ B<0 \to \textrm{saddle cycle } \mathcal{C}, \textrm{ attracting } x\textrm{-}y \textrm{ plane}$$

$$A>0,\ B>0 \to \textrm{instable cycle } \mathcal{C}$$

To embed this system in the scope of DynSys3D, it has to be defined on the basis of Cartesian coordinates:

$$r\cdot{}\cos\phi$$ x =

$$r\cdot{}\sin\phi$$ y =

The dynamical system in terms of Cartesian coordinates is then given as:

$$\dot{x} a\cdot{}x - y,\quad{}a=A\cdot{}(r-1/r),\ r=\sqrt{x^2+y^2}$$ =

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

$$\dot{z} b\cdot{}z,\quad{}b=B$$ =

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