On the homotopy classification of positively homogeneous functions of three variables

In this paper, we study the problem of homotopy classification of the set $\mathcal{F}$ of positively homogeneous smooth functions in three variables whose gradients do not vanish at nonzero points. This problem is of interest in the study of periodic and bounded solutions of systems of ordinary differential equations with the main positive homogeneous nonlinearity. The subset $\mathcal{F}_0\subset\mathcal{F}$ is presented and for any function $g(x)\in\mathcal{F}_0$, a formula for calculating the rotation $\gamma (\nabla g)$ of its gradient $\nabla g(x)$ on the boundary of the unit ball $|x| <1$ is derived. It is proved that any function from $\mathcal{F}$ is homotopic to some function from $\mathcal{F}_0$.