NLCYC1

Another test case for the Poincaré section techniques is NLCYC1, which is a dynamical system that exhibits one cycle with a non-linear Poincaré map.

First we specify a linear 2D system with a critical point in the origin and characteristic directions coinciding with the axes of the system:

$$\dot{\alpha} A\cdot{}\alpha$$ =

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

Next we apply a non-linear transformation to variable $\alpha$:  $u=\alpha+\beta^2$, and $v=\beta$. This transformation yields another 2D dynamical system as follows:

$$\dot{u} A\cdot{}u + (2B-A)\cdot{}v^2$$ =

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

We now place the entire system into the 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:

$$\dot{u'}/u' A\cdot{}\ln{}u' + (2B-A)\cdot{}v'^2$$ =

$$\dot{v'} B\cdot{}v'$$ =

Finally we transform the system into 3D by rotating it around the z-axis (r=u', $\dot{\phi}=C$, and z=v'). This yields the final system as follows:

$$\dot{x} x\cdot{}(A\cdot{}\ln{}r + (2B-A)\cdot{}z^2) - C\cdot{}y,\quad{}r = \sqrt{x^2 + y^2}$$ =

$$\dot{y} y\cdot{}(A\cdot{}\ln{}r + (2B-A)\cdot{}z^2) + C\cdot{}x$$ =

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

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