Marc Georgi

Evolution of convex lens-shaped networks under the curve shortening flow

We consider convex symmetric lens-shaped networks in R2 that evolve under the curve shortening flow. We show that the enclosed convex domain shrinks to a point in finite time. Furthermore, after appropriate rescaling the evolving networks converge to a self-similarly shrinking network, which we prove to be unique in an appropriate class. We also include a classification result for some self-similarly shrinking networks.

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.

Bifurcations from homoclinic orbits to non-hyperbolic equilibria in reversible lattice differential equations

Starting with a lattice differential equation (LDE), the study of travelling wave solutions leads to a forward–backward delay equation. Structures which are present in the LDE typically are inherited by the travelling wave equation. In this paper we are interested in a situation where the travelling wave equation is reversible and possesses a symmetric homoclinic solution. Moreover, we consider the case that the asymptotic steady state possesses exactly two purely imaginary eigenvalues ±iω, ω ≠ 0. As a consequence, a family of small periodic solutions exist near the steady state. It is the aim of this paper to analyse this bifurcation under generic assumptions and to exploit the underlying reversibility of the equation.As one of the main results we find all homoclinic orbits to the centre manifold, which approach a periodic orbit in forward and backward time.

Structural stability of travelling waves in an integrodifferential equation

In this article, we study the structural stability of travelling waves of an integrodifferential equation, which can be viewed as the non-local analogon of the reaction-diffusion equation [udot] = uxx + f(u). More precisely, we are interested in the question whether a travelling wave solution persists under small perturbations of the equation. Since the travelling wave equation is a functional differential equation of mixed type, a deeper understanding of the intersection of stable and unstable manifold of the steady state in mixed type equations turns out to be crucial. As one of the main results, we prove the existence of stable and unstable manifolds for general functional differential equations. We apply our results to the one-dimensional equation of elasticity with non-local energy. In particular, we prove that a travelling wave is structural stable if and only if the underlying shock wave is compressive.

Interaction of homoclinic solutions and Hopf points in functional differential equations of mixed type

In this article we study a homoclinic bifurcation in a general functional differential equation of mixed type. More precisely, we investigate the case when the asymptotic steady state of a homoclinic solution undergoes a Hopf bifurcation. Bifurcations of this kind are hard to analyse due to the lack of Fredholm properties. In particular, a straightforward application of a Lyapunov-Schmidt reduction is not possible. As one of the main results we prove the existence of center stable and center unstable manifolds of steady states near homoclinic orbits. With their help, we can analyse the bifurcation scenario similar to the ODE-case and can show the existence of solutions which bifurcate near the homoclinic orbit, are decaying in one direction and oscillatory in the other direction. These solutions can be visualized as an interaction of the homoclinic orbit and small periodic solutions, which exist on account of the Hopf bifurcation, for exactly one asymptotic direction t → ∞ or t → −∞.

Bifurcations from localised steady states to generalised breather solutions in the Klein-Gordon lattice

We consider an infinite chain of particles linearly coupled to their nearest neighbours and look for time-periodic, spatially almost localised solutions (generalised breathers). As a starting point, we consider a time-independent breather that induces a transverse homoclinic solution of a two-dimensional recurrence relation. We can prove the existence of any finite number of (generalised) breathers which bifurcate from the time-independent breather solution at low frequency. One of the main motivations of this paper is to provide a set up, where the existence-proof of chaotic behaviour near (generalised) breathers becomes accessible to analytical methods.