Kevin N. Webster

Ancient Dynamics in Bianchi Models: Approach to Periodic Cycles

We consider cosmological models of Bianchi type. In particular, we are interested in the α-limit dynamics near the Kasner circle of equilibria for Bianchi classes VIII and IX. They correspond to cosmological models close to the big-bang singularity.We prove the existence of a codimension-one family of solutions that limit, for t → −∞, onto a heteroclinic 3-cycle to the Kasner circle of equilibria. The theory extends to arbitrary heteroclinic chains that are uniformly bounded away from the three critical Taub points on the Kasner circle, in particular to all closed heteroclinic cycles of the Kasner map.

Asymptotic analysis of the Michelson system

We consider the dynamical system xttt = c2−½x2−xt for the parameter c close to zero. We perform a multiple timescale analysis to provide analytic forms for all bounded solutions of the formal normal form in the phase space, in a neighbourhood of the origin (x,c) = (0, 0). These take the form of Jacobi elliptic functions describing periodic and quasi-periodic solutions, and hyperbolic functions that describe heteroclinic connections. A comparison between these approximate analytical results and numerical simulations of the unperturbed system shows excellent correspondence.

Approximation of reachable sets using optimal control and support vector machines

We propose and discuss a new computational method for the numerical approximation of reachable sets for nonlinear control systems. It is based on the support vector machine algorithm and represents the set approximation as a sublevel set of a function chosen in a reproducing kernel Hilbert space. In some sense, the method can be considered as an extension to the optimal control algorithm approach recently developed by Baier, Gerdts and Xausa. The convergence of the method is illustrated numerically for selected examples.

Shift dynamics near non-elementary T-points with real eigenvalues

Abstract We consider non-elementary T-points in reversible systems in . We assume that the leading eigenvalues are real. We prove the existence of shift dynamics in the unfolding of this T-point. Furthermore, we study local bifurcations of symmetric periodic orbits occurring in the process of dissolution of the chaotic dynamics.

NORMAL FORMS, QUASI-INVARIANT MANIFOLDS, AND BIFURCATIONS OF NONLINEAR DIFFERENCE-ALGEBRAIC EQUATIONS

We study the existence of quasi-invariant manifolds in a neighborhood of a fixed point of the difference-algebraic equation (

Bounding the first exit from the basin: Independence times and finite-time basin stability

\Delta

A reinterpretation of set differential equations as differential equations in a Banach space

AE)

A Hopf bifurcation theorem for singular differential-algebraic equations

F(z_n,z_{n+1})=0

Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in R3

, where

Approximation of Lyapunov Functions from Noisy Data

F:\mathbb{R}^{2m} \to \mathbb{R}^m