A flow on \(S^2\) presenting the ball as its minimal set (Q2033790)

The main result of the paper is a construction of a piecewise smooth vector field (the so-called Filippov's vector field) tangent to the unit sphere \(S^2\) such that \(S^2\) itself is a non-trivial minimal set. The construction is based on switching two planar vector fields associated with the following ordinary differential equations: \[ \left\{\begin{array}{l} \dot{x}=2,\\ \dot{y}=y(x^3-3x), \end{array}\right. \left\{\begin{array}{l} \dot{x}=-1,\\ \dot{y}=x(x/2-y), \end{array}\right. \] and on using stereographic projections.