### 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

$\lambda =$ You must enter a decimal number between $-1$ and $1$.

Projection of an orbit on the Poincaré sphere

## Bibliography

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