### Poincaré sphere

The Poincaré sphere can be used to study the behavior at infinity of some dynamical systems [Perko (2001)]. The projection equations defining the coordinates \((x,~y)\) of a generic point of the plane, in terms of the coordinates \((X,~Y,~Z)\) of a corresponding point on the sphere, read:

\begin{equation*}
\begin{cases}
x = \dfrac{X}{Z}\\
y = \dfrac{Y}{Z}
\end{cases}.
\end{equation*}

The inverse transformation is given by:
\begin{equation*}
\begin{cases}
X = \dfrac{x}{\sqrt{1+x^2+y^2}}\\
Y = \dfrac{y}{\sqrt{1+x^2+y^2}}\\
Z = \dfrac{1}{\sqrt{1+x^2+y^2}}
\end{cases}.
\end{equation*}

You can visualize the orbit of the following system, that exhibits a vertical Hopf bifurcation at
\(\lambda = 0 \):
\begin{equation*}
\begin{cases}
\dot{x} = \lambda x - y \\
\dot{y} = x + \lambda y
\end{cases}.
\end{equation*}

For \(\lambda > 0 \), the orbit is attracted to the equator.
For \(\lambda < 0 \), it is attracted to the north pole.
### Simulation

*Projection of an orbit on the Poincaré sphere*

## Bibliography

L. Perko.
*Differential equations and dynamical systems*.
Springer, New York, 3rd edition, 2001. ↩