New structurally unstable families of planar vector fields

construction of vector fields with positive lyapunov exponents

in this thesis our aim is to construct vector field in r3 for which the corresponding one-dimensional maps have certain discontinuities. two kinds of vector fields are considered, the first the lorenz vector field, and the second originally introced here. the latter have chaotic behavior and motivate a class of one-parameter families of maps which have positive lyapunov exponents for an open in...

New almost-planar crossing-critical graph families

We show that, for all choices of integers k > 2 and m, there are simple 3-connected k-crossing-critical graphs containing more than m vertices of each even degree ≤ 2k − 2. This construction answers one half of a question raised by Bokal, while the other half asking analogously about vertices of odd degrees at least 5 in crossing-critical graphs remains open. Furthermore, our constructed graphs...

Linearization of families of germs of hyperbolic vector fields

Families of Vector Fields which Generate the Group of Diffeomorphisms

Given a compact manifold M , we prove that any bracket generating and invariant under multiplication on smooth functions family of vector fields on M generates the connected component of unit of the group DiffM . Let M be a smooth n-dimensional compact manifold, VecM the space of smooth vector fields on M and Diff0M the group of isotopic to the identity diffeomorphisms of M . Given f ∈ VecM , w...

Recurrent switching rules for pairs of linear planar vector fields

Given a switched system formed by a pair of (not asymptotically stable) linear vector fields of R, we define a special class of hybrid feedback laws (called recurrent switching rules) whose structure depends on some conic regions with nonempty interior. We prove that if the system is asymptotically controllable and radially controllable, then it is robustly stabilized by any recurrent switching...