On two-dimensional integral varieties of a~class of discontinuous Hamiltonian systems

V. F. Borisov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 6 (2000) no. 2, pp. 357-377.

Résumé

We consider the following discontinuous Hamiltonian system \begin{gather*} \dot y=I\operatorname{grad}H(y), \\ H(y)=H_0(y)+u H_1(y),\quad u=\operatorname{sgn}H_1(y),\quad I=\begin{pmatrix} 0 -E \\ E 0 \end{pmatrix}. \end{gather*} Here $E$ is the unit $(n\times n)$-matrix, $y\in\mathbb R^{2n}$. Under general assumptions, we prove that a vicinity of a singular extremal of order $q$ ($2\le q\le n$) contains $[q/2]$ integral varieties with chattering trajectories. That means that the trajectories enter into the singular extremal at a finite instant with an infinite number of intersections with the surface of discontinuity (Fuller's phenomenon).