Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation

A nonlinear singular perturbation problem

Let F (uε) + ε(uε − w) = 0 (1) where F is a nonlinear operator in a Hilbert space H, w ∈ H is an element, and ε > 0 is a parameter. Assume that F (y) = 0, and F ′(y) is not a boundedly invertible operator. Sufficient conditions are given for the existence of the solution to (1) and for the convergence limε→0 ‖uε−y‖ = 0. An example of applications is considered. In this example F is a nonlinear ...

Bifurcations and chaos in converters . Discontinuous vector fields and singular Poincaré maps

Basic geometrical and topological features are described for discontinuous systems modelling power converters. The global Poincaré map considered arises naturally from the sampling process in the oscillatory forced system. It is shown that this map belongs to the class of two-dimensional invertible continuous but only piecewise-smooth maps. It also contains a singular point at which critical cu...

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...

Singular Vector Perturbation Under Gaussian Noise

We perform a non-asymptotic analysis on the singular vector distribution under Gaussian noise. In particular, we provide sufficient conditions on a matrix for its first few singular vectors to have near normal distribution. Our result can be used to facilitate the error analysis in linear dimension reduction.

Controllability and stabilizability of bimodal systems with discontinuous vector fields