# Dr Claudia Wulff

PhD
+44 (0)1483 682630
10 AA 04

### Research interests

#### 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.

### Highlights

Full details can be found on my personal website.

### Publications

Lamb JSW, Melbourne I, Wulff C (2003) Bifurcation from periodic solutions with spatiotemporal symmetry, including resonances and mode interactions, JOURNAL OF DIFFERENTIAL EQUATIONS 191 (2) pp. 377-407 ACADEMIC PRESS INC ELSEVIER SCIENCE
Wulff C (2008) A Hamiltonian Analogue of the Meandering Transition, SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 7 (4) pp. 1213-1246 SIAM PUBLICATIONS
Oliver M, Wulff C (2010) A-stable Runge-Kutta methods for semilinear evolution equations,
We consider semilinear evolution equations for which the linear part
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.
Wulff Claudia, Evans C (2017) Numerische Mathematik Springer Verlag
We study semilinear evolution equations $\frac {\d U}{\d t}=AU+B(U)$ posed on a Hilbert space $\kY$, where $A$ is normal and generates a strongly continuous semigroup, $B$ is a smooth nonlinearity from $\kY_\ell = D(A^\ell)$ to itself, and $\ell \in I \subseteq [0,L]$, $L \geq 0$, $0,L \in I$. In particular the one-dimensional semilinear wave equation and nonlinear Schr\"odinger equation with periodic, Neumann and Dirichlet boundary conditions fit into this framework. We discretize the evolution equation with an A-stable Runge-Kutta method in time, retaining continuous space, and prove convergence of order $O(h^{p\ell/(p+1)})$ for non-smooth initial data $U^0\in \kY_\ell$, where $\ell\leq p+1$, for a method of classical order $p$, extending a result by Brenner and Thom\'ee for linear systems. Our approach is to project the semiflow and numerical method to spectral Galerkin approximations, and to balance the projection error with the error of the time discretization of the projected system. Numerical experiments suggest that our estimates are sharp.
Wulff C (2006) JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS 12 (11) pp. 1179-1191 TAYLOR & FRANCIS LTD
Wulff C (2014) SIAM J. Applied Dynamical Systems 13 (2) pp. 722-757 SIAM
Relative equilibria and relative periodic orbits (RPOs) are ubiquitous in symmetric Hamiltonian systems and occur for example in celestial mechanics, molecular dynamics and rigid body motion. Relative equilibria are equilibria and RPOs are periodic orbits in the symmetry reduced system. Relative Lyapunov centre bifurcations are bifurcations of relative periodic orbits from relative equilibria corresponding to Lyapunov centre bifurcations of the symmetry reduced dynamics. In this paper we first prove a relative Lyapunov centre theorem by combining recent results on persistence of RPOs in Hamiltonian systems with a symmetric Lyapunov centre theorem of Montaldi et al. We then develop numerical methods for the detection of relative Lyapunov centre bifurcations along branches of RPOs and for their computation. We apply our methods to Lagrangian relative equilibria of the N body problem.
Wulff C, Schebesch A (2006) SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 5 (3) pp. 435-475 SIAM PUBLICATIONS
Oliver M, Wulff C (2012) Journal of Functional Analysis 263 (7) pp. 1981-2023
We consider semilinear evolution equations for which the linear part 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 Schrödinger equation. © 2012 Elsevier Inc.
Wulff C, Schilder F (2009) SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 8 (3) pp. 931-966 SIAM PUBLICATIONS
Wulff C, Correa. D, Piqueira, J (2014) Symmetric bifurcation analysis of synchronous states of time-delayed coupled Phase-Locked Loop oscillators, In: Symmetric bifurcation analysis of synchronous states of time-delayed coupled Phase-Locked Loop oscillators arxiv
In recent years there has been an increasing interest in studying time-delayed coupled networks of oscillators since these occur in many real life applications. In many cases symmetry patterns can emerge in these networks, as a consequence a part of the system might repeat itself, and properties of this symmetric subsystem are representative of the whole dynamics. In this paper an analysis of the second order N-node time-delay
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.
Ferruzzo Correa DP, Wulff C, Piqueira JRC (2015) Communications in Nonlinear Science and Numerical Simulation 22 (1-3) pp. 793-820 Elsevier
In recent years there has been an increasing interest in studying time-delayed coupled networks of oscillators since these occur in many real life applications. In many cases symmetry patterns can emerge in these networks, as a consequence a part of the system might repeat itself, and properties of this subsystem are representative of the dynamics on the whole phase space. In this paper an analysis of the second order N-node time-delay fully connected network is presented which is based on previous work: synchronous states in time-delay coupled periodic oscillators: a stability criterion. Correa and Piqueira (2013), 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 Hopf bifurcations. Three different models are used to analyze the network dynamics, namely, the full-phase, the phase, and the phase-difference model. We determine a finite set of frequencies É, that might correspond to Hopf bifurcations in each case for critical values of the delay. The Sn map is used to actually find Hopf bifurcations along with numerical calculations using the Lambert W function. Numerical simulations are used in order to confirm 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. © 2014 Elsevier B.V. All rights reserved.
Wulff C (2003) JOURNAL OF GEOMETRY AND PHYSICS 48 (2-3) pp. 309-338 ELSEVIER SCIENCE BV
Wulff C (2014) Proc. Royal Soc. Edinb. A 144 (3) pp. 603-636 Cambridge Journals
We consider semilinear evolution equations for which the linear part is 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.
Lamb JSW, Melbourne I, Wulff C (2006) Hopf bifurcation from relative periodic solutions; secondary bifurcations from meandering spirals, JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS 12 (11) pp. 1127-1145 TAYLOR & FRANCIS LTD
Wulff C, Schilder F, Schebesch A (2010) SYMPERCON (Continuation software for symmetric periodic orbits) User Manual, SourceForge.net
Wulff C, Schebesch A (2008) Journal of Nonlinear Science 18 (4) pp. 343-390
The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years, there has been rapid progress in the development of a bifurcation theory for dynamical systems with structure, such as symmetry or symplecticity. But as yet, there are few results on the numerical computation of those bifurcations. The methods we present in this paper are a first step toward a systematic numerical analysis of generic bifurcations of Hamiltonian symmetric periodic orbits and relative periodic orbits (RPOs). First, we show how to numerically exploit spatio-temporal symmetries of Hamiltonian periodic orbits. Then we describe a general method for the numerical computation of RPOs persisting from periodic orbits in a symmetry breaking bifurcation. Finally, we present an algorithm for the numerical continuation of non-degenerate Hamiltonian relative periodic orbits with regular drift-momentum pair. Our path following algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincaré section and a tangential continuation method with implicit reparametrization. We apply our methods to continue the famous figure eight choreography of the three-body system. We find a relative period doubling bifurcation of the planar rotating eight family and compute the rotating choreographies bifurcating from it. © 2008 Springer Science+Business Media, LLC.
Oliver M, Wulff C (2010) Stability under Galerkin truncation of A-stable Runge--Kutta
discretizations in time,
We consider semilinear evolution equations for which the linear part is
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.
Oliver M, West M, Wulff C (2004) Approximate momentum conservation for spatial semidiscretizations of semilinear wave equations, NUMERISCHE MATHEMATIK 97 (3) pp. 493-535 SPRINGER
Kristiansen KU, Wulff Claudia (2012) arXiv arXiv
In this paper we prove the existence of an almost invariant symplectic slow manifold for analytic Hamiltonian slow-fast systems with finitely many slow degrees of freedom for which the error field is exponentially small. We allow for infinitely many fast degrees of freedom. The method we use is motivated by a paper of MacKay from 2004. The method does not notice resonances, and therefore we do not pose any restrictions on the motion normal to the slow manifold other than it being fast and analytic. We also present a stability result and obtain a generalization of a result of Gelfreich and Lerman on an invariant slow manifold to (finitely) many fast degrees of freedom.
Patrick G, Roberts M, Wulff C (2008) NONLINEARITY 21 (2) pp. 325-352 IOP PUBLISHING LTD

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.

Patrick G, Roberts M, Wulff C (2004) ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS 174 (3) pp. 301-344 SPRINGER
We develop a general stability theory for equilibrium points of Poisson dynamical systems and relative equilibria of Hamiltonian systems with symmetries, including several generalisations of the Energy-Casimir and Energy-Momentum Methods. Using a topological generalisation of Lyapunov?s result that an extremal critical point of a conserved quantity is stable, we show that a Poisson equilibrium is stable if it is an isolated point in the intersection of a level set of a conserved function with a subset of the phase space that is related to the topology of the symplectic leaf space at that point. This criterion is applied to generalise the energy-momentum method to Hamiltonian systems which are invariant under non-compact symmetry groups for which the coadjoint orbit space is not Hausdorff. We also show that a G-stable relative equilibrium satisfies the stronger condition of being A-stable, where A is a specific group-theoretically defined subset of G which contains the momentum isotropy subgroup of the relative equilibrium. The results are illustrated by an application to the stability of a rigid body in an ideal irrotational fluid.
Wulff Claudia, Oliver M (2016) PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS 146 (6) pp. 1265-1301 ROYAL SOC EDINBURGH
We prove that a class of A-stable symplectic Runge--Kutta time semidiscretizations (including the Gauss--Legendre methods) applied to a class of semilinear Hamiltonian PDEs which are well-posed on spaces of analytic functions with analytic initial data can be embedded into a modified Hamiltonian flow up to an exponentially small error. As a consequence, such time-semidiscretizations conserve the modified Hamiltonian up to an exponentially small error. The modified Hamiltonian is O(h^p)-close to the original energy where p is the order of the method and h the time step-size. Examples of such systems are the semilinear wave equation or the nonlinear Schr\"odinger equation with analytic nonlinearity and periodic boundary conditions. Standard Hamiltonian interpolation results do not apply here because of the occurrence of unbounded operators in the construction of the modified vector field. This loss of regularity in the construction can be taken care of by projecting the PDE to a subspace where the operators occurring in the evolution equation are bounded and by coupling the number of excited modes as well as the number of terms in the expansion of the modified vector field with the step size. This way we obtain exponential estimates of the form O(\exp(-c/h^{1/(1+q)})) with c>0 and q \geq 0; for the semilinear wave equation, q=1, and for the nonlinear Schr\"odinger equation, q=2. We give an example which shows that analyticity of the initial data is necessary to obtain exponential estimates.
Wulff Claudia, Kristiansen K (2016) Journal of Differential Equations 261 (1) pp. 56-101 Elsevier
In this paper we prove the existence of an almost invariant symplectic slow manifold for analytic Hamiltonian slow-fast systems with finitely many slow degrees of freedom for which the error field is exponentially small. We allow for infinitely many fast degrees of freedom. The method we use is motivated by a paper of MacKay from 2004. The method does not notice resonances, and therefore we do not pose any restrictions on the motion normal to the slow manifold other than it being fast and analytic. We also present a stability result and obtain a generalization of a result of Gelfreich and Lerman on an invariant slow manifold to (finitely) many fast degrees of freedom.