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.