# Dr Claudia Wulff

### Biography

I have been at the University of Surrey since 2002. I got my PhD in 1996 at Freie Universitaet Berlin.

### Research

### Research interests

- Dynamical Systems with Symmetry
- Hamiltonian systems
- Numerics of Dynamical Systems
- Nonlinear PDEs and Pattern Formation.

#### Former grants (2007-2009)

- EPSRC grant "Symmetric Hamiltonian systems: bifurcation theory and numerics", 178000 pounds, research fellow Dr. Frank Schilder* (2007-2008)
- Leverhulme fellowship "Exponential error estimates for Hamiltonian PDEs and their discretizations", 30000 pounds.

### My teaching

In the academic year 2018/19 I teach:

- Stochastic Processes MAT2003. All information about this module can be found on SurreyLearn.
- Symplectic Numerical Methods MATM051. All information about this module can be found on SurreyLearn.
- First year tutorials, see your personal timetable for the times and dates.

### My publications

### Publications

generates a strongly continuous semigroup and the nonlinear part is

sufficiently smooth on a scale of Hilbert spaces. In this setting, we prove the

existence of solutions which are temporally smooth in the norm of the lowest

rung of the scale for an open set of initial data on the highest rung of the

scale. Under the same assumptions, we prove that a class of implicit,

$A$-stable Runge--Kutta semidiscretizations in time of such equations are

smooth as maps from open subsets of the highest rung into the lowest rung of

the scale. Under the additional assumption that the linear part of the

evolution equation is normal or sectorial, we prove full order convergence of

the semidiscretization in time for initial data on open sets. Our results

apply, in particular, to the semilinear wave equation and to the nonlinear

Schr\"odinger equation.

fully connected network is made based on previous work by Correa and

Piqueira [8] for a 2-node network. This study is carried out using symmetry groups. We show the existence of multiple eigenvalues forced by

symmetry, as well as the existence of steady-state and Hopf bifurcations

(in [8] only Hopf bifurcations were found). Three diff erent models are used

to analyze the network dynamics, namely, the full-phase, the phase, and

the phase diff erence model. We determine a nite set of frequencies

that might correspond to Hopf bifurcations in each case for critical values of the delay. The Sn map proposed in [10] is used to actually find Hopf bifurcations along with numerical calculations using the Lambert W function. Numerical simulations are used in order to con firm the analytical results. Although we restrict attention to second order nodes, the results could be extended to higher order networks provided the time delay in

the connections between nodes remains equal.

discretizations in time,

normal and generates a strongly continuous semigroup and the nonlinear part is

sufficiently smooth on a scale of Hilbert spaces. We approximate their semiflow

by an implicit, A-stable Runge--Kutta discretization in time and a spectral

Galerkin truncation in space. We show regularity of the Galerkin-truncated

semiflow and its time-discretization on open sets of initial values with bounds

that are uniform in the spatial resolution and the initial value. We also prove

convergence of the space-time discretization without any condition that couples

the time step to the spatial resolution. Then we estimate the Galerkin

truncation error for the semiflow of the evolution equation, its Runge--Kutta

discretization, and their respective derivatives, showing how the order of the

Galerkin truncation error depends on the smoothness of the initial data. Our

results apply, in particular, to the semilinear wave equation and to the

nonlinear Schr\"odinger equation.

In the presence of noncompact symmetry, the stability of relative equilibria under momentum-preserving perturbations does not generally imply robust stability under momentum-changing perturbations. For axisymmetric relative equilibria of Hamiltonian systems with Euclidean symmetry, we investigate different mechanisms of stability: stability by energy?momentum confinement, KAM, and Nekhoroshev stability, and we explain the transitions between them. We apply our results to the Kirchhoff model for the motion of an axisymmetric underwater vehicle, and we numerically study dissipation induced instability of KAM stable relative equilibria for this system.