On piecewise smooth vector fields tangent to nested tori (Q305417)

The authors start with a piecewise smooth vector field on \(\mathbb R^3\) of the form NEWLINE\[NEWLINEZ_0 = ((\mathrm{sgn} z)x, (\mathrm{sgn} z)y, f(x^2 + y^2))NEWLINE\]NEWLINE with switching set the \((x, y)\)-plane that is tangent to a foliation of \(\mathbb R^3\) whose leaves are topological 2-tori except for two topological circles, tangent to all planes that contain the \(z\)-axis, and all of whose orbits are topological circles. They prove the existence, for each \(k \in \mathbb Z^+ \cup \{ \infty \}\), of a one-parameter family of piecewise smooth vector fields \(Z_\varepsilon^k = Z_\varepsilon\) that satisfies a number of properties, among them: (a) \(Z_\varepsilon \to Z_0\) as \(\varepsilon \to 0\); (b) \(Z_\varepsilon\) has exactly \(k\) hyperbolic nested limit cycles in each connected component of a vertical plane through the \(z\)-axis, minus the \(z\)-axis; (c) \(Z_\varepsilon\) has exactly \(k\) hyperbolic nested invariant topological tori; and (d) a particular circle \(x^2 + y^2 = r^2\) in the plane \(z = 0\) is an attractor for all trajectories of \(Z_\varepsilon\).NEWLINENEWLINEThey prove a similar theorem for starting system \(Z_0^\eta\) in which a small rotation \(\eta\) has been introduced for \(z< 0\).